A291845 - OEIS

%I #8 Sep 03 2017 16:39:10

%S 1,1,1,1,1,4,5,4,1,1,9,26,33,26,9,1,1,16,90,224,283,224,90,16,1,1,25,

%T 235,1050,2389,2995,2389,1050,235,25,1,1,36,511,3660,14174,30324,

%U 37723,30324,14174,3660,511,36,1,1,49,980,10339,62265,218246,446109,551047,446109,218246,62265,10339,980,49,1,1,64,1716,25088,218330,1162560,3782064,7460928,9157923,7460928,3782064,1162560,218330,25088,1716,64,1,1,81,2805,54324,646542,4899258,23763914,72918576,139775763,170606547,139775763,72918576,23763914,4899258,646542,54324,2805,81,1

%N Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-1} (1 + (2*k+1)*x + x^2) for n>0 with a single '1' in row 0.

%C Row sums yield the odd double factorials A001147.

%C Central terms in rows form A291846.

%C Another diagonal forms A291847.

%C Antidiagonal sums yield A291848.

%H Paul D. Hanna, <a href="/A291845/b291845.txt">Table of n, a(n) for n = 0..1680 of rows 0..40 of this triangle in flattened form.</a>

%e This irregular triangle begins:

%e 1;

%e 1, 1, 1;

%e 1, 4, 5, 4, 1;

%e 1, 9, 26, 33, 26, 9, 1;

%e 1, 16, 90, 224, 283, 224, 90, 16, 1;

%e 1, 25, 235, 1050, 2389, 2995, 2389, 1050, 235, 25, 1;

%e 1, 36, 511, 3660, 14174, 30324, 37723, 30324, 14174, 3660, 511, 36, 1;

%e 1, 49, 980, 10339, 62265, 218246, 446109, 551047, 446109, 218246, 62265, 10339, 980, 49, 1;

%e 1, 64, 1716, 25088, 218330, 1162560, 3782064, 7460928, 9157923, 7460928, 3782064, 1162560, 218330, 25088, 1716, 64, 1;

%e 1, 81, 2805, 54324, 646542, 4899258, 23763914, 72918576, 139775763, 170606547, 139775763, 72918576, 23763914, 4899258, 646542, 54324, 2805, 81, 1;

%e 1, 100, 4345, 107700, 1681503, 17237880, 117496358, 529332200, 1548992621, 2899264620, 3521075919, 2899264620, 1548992621, 529332200, 117496358, 17237880, 1681503, 107700, 4345, 100, 1; ...

%o (PARI) {T(n, k)=polcoeff(prod(j=0, n-1, 1 + (2*j+1)*x + x^2), k)}

%o {for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))}

%Y Cf. A291846, A291847, A291848, A201949, A001147 (row sums).

%K nonn,tabf

%O 0,6

%A _Paul D. Hanna_, Sep 03 2017