A114172 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

A114172

Triangle, read by rows, where the g.f. of column n, C_n(x), equals the g.f. of row n, R_n(x), divided by (1-x)^(n+1), for n>=0; e.g., C_n(x) = R_n(x)/(1-x)^(n+1).

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 16, 9, 1, 1, 9, 31, 36, 12, 1, 1, 11, 51, 95, 66, 15, 1, 1, 13, 76, 199, 229, 106, 18, 1, 1, 15, 106, 361, 601, 467, 156, 21, 1, 1, 17, 141, 594, 1316, 1509, 844, 216, 24, 1, 1, 19, 181, 911, 2542, 3951, 3293, 1395, 286, 27, 1, 1, 21, 226

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Table of n, a(n) for n=0..68.

EXAMPLE

Triangle begins:

1,1;

1,3,1;

1,5,6,1;

1,7,16,9,1;

1,9,31,36,12,1;

1,11,51,95,66,15,1;

1,13,76,199,229,106,18,1;

1,15,106,361,601,467,156,21,1;

1,17,141,594,1316,1509,844,216,24,1;

1,19,181,911,2542,3951,3293,1395,286,27,1;

1,21,226,1325,4481,8910,10193,6447,2155,366,30,1; ...

Where g.f. for columns is formed from g.f. of rows:

GF(column 2) = (1 + 3*x + 1*x^2)/(1-x)^3

= 1 + 6*x + 16*x^2 + 31*x^3 + 51*x^4 + 76*x^5 +...

GF(column 3) = (1 + 5*x + 6*x^2 + 1*x^3)/(1-x)^4

= 1 + 9*x + 36*x^2 + 95*x^3 + 199*x^4 + 361*x^5 +...

GF(column 4) = (1 + 7*x + 16*x^2 + 9*x^3 + 1*x^4)/(1-x)^5

= 1 + 12*x + 66*x^2 + 229*x^3 + 601*x^4 + 1316*x^5 +...

PROG

(PARI)

{T(n, k)=if(n<k||k<0, 0, if(n==k||k==0, 1, polcoeff(sum(j=0, k, T(k, j)*x^j)/(1-x+x*O(x^(n-k)))^(k+1), n-k)))}

CROSSREFS

Cf. A114173 (row sums), A114174 (central terms), A114175 (row sums-square).

Sequence in context: A202672 A054142 A076756 * A271942 A121522 A294582

Adjacent sequences: A114169 A114170 A114171 * A114173 A114174 A114175

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Nov 15 2005

STATUS

approved