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A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2*x + (k/2)*x^2), n >= 0, k >= 0.

%I A376826 #9 Oct 07 2024 18:35:50
%S A376826 1,1,2,1,2,4,1,2,5,8,1,2,6,14,16,1,2,7,20,43,32,1,2,8,26,76,142,64,1,
%T A376826 2,9,32,115,312,499,128,1,2,10,38,160,542,1384,1850,256,1,2,11,44,211,
%U A376826 832,2809,6512,7193,512,1,2,12,50,268,1182,4864,15374,32400,29186,1024
%N A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2*x + (k/2)*x^2), n >= 0, k >= 0.
%H A376826 Andrew Howroyd, <a href="/A376826/b376826.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H A376826 Arvind Ayyer, Hiranya Kishore Dey and Digjoy Paul, <a href="https://arxiv.org/abs/2406.06036">How large is the character degree sum compared to the character table sum for a finite group?</a>, arXiv preprint arXiv:2406.06036, [math.RT], 2024.
%F A376826 E.g.f. of column k: exp(2*x + k*x^2/2).
%F A376826 Column k is the binomial transform of column k of A359762.
%F A376826 T(n,k) = Sum_{i=0..floor(n/2)} binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i-1)!!.
%F A376826 T(n,k) = Sum_{i=0..floor(n/2)} 2^(n-3*i) * k^i * n! / ((n-2*i)! * i!).
%e A376826 Array begins:
%e A376826 ======================================================
%e A376826 n\k |   0    1    2     3     4     5     6      7 ...
%e A376826 ----+-------------------------------------------------
%e A376826   0 |   1    1    1     1     1     1     1      1 ...
%e A376826   1 |   2    2    2     2     2     2     2      2 ...
%e A376826   2 |   4    5    6     7     8     9    10     11 ...
%e A376826   3 |   8   14   20    26    32    38    44     50 ...
%e A376826   4 |  16   43   76   115   160   211   268    331 ...
%e A376826   5 |  32  142  312   542   832  1182  1592   2062 ...
%e A376826   6 |  64  499 1384  2809  4864  7639 11224  15709 ...
%e A376826   7 | 128 1850 6512 15374 29696 50738 79760 118022 ...
%e A376826      ...
%o A376826 (PARI) T(n,k) = {sum(i=0, n\2, binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i)!/(2^i*i!))}
%Y A376826 Columns 0..5 are A000079, A005425, A000898, A202830, A193778, A202832.
%Y A376826 Cf. A359762, A373625.
%K A376826 nonn,tabl
%O A376826 0,3
%A A376826 _Andrew Howroyd_, Oct 07 2024