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A292627 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*BesselI(0,2*x).

%I A292627 #40 Aug 18 2025 09:34:46
%S A292627 1,1,0,1,1,2,1,2,3,0,1,3,6,7,6,1,4,11,20,19,0,1,5,18,45,70,51,20,1,6,
%T A292627 27,88,195,252,141,0,1,7,38,155,454,873,924,393,70,1,8,51,252,931,
%U A292627 2424,3989,3432,1107,0,1,9,66,385,1734,5775,13236,18483,12870,3139,252,1,10,83,560,2995,12276,36645,73392,86515,48620,8953,0
%N A292627 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*BesselI(0,2*x).
%C A292627 A(n,k) is the k-th binomial transform of A126869 evaluated at n.
%H A292627 Seiichi Manyama, <a href="/A292627/b292627.txt">Antidiagonals n = 0..139, flattened</a>
%H A292627 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A292627 O.g.f. of column k: 1/sqrt( (1 - (k-2)*x)*(1 - (k+2)*x) ).
%F A292627 E.g.f. of column k: exp(k*x)*BesselI(0,2*x).
%F A292627 From _Seiichi Manyama_, May 01 2019: (Start)
%F A292627 A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + x^2)^n.
%F A292627 A(n,k) = Sum_{j=0..n} (k-2)^(n-j) * binomial(n,j) * binomial(2*j,j).
%F A292627 A(n,k) = Sum_{j=0..n} (k+2)^(n-j) * (-1)^j * binomial(n,j) * binomial(2*j,j).
%F A292627 n * A(n,k) = k * (2*n-1) * A(n-1,k) - (k^2-4) * (n-1) * A(n-2,k). (End)
%F A292627 A(n,k) = Sum_{j=0..floor(n/2)} k^(n-2*j) * binomial(n,2*j) * binomial(2*j,j). - _Seiichi Manyama_, May 04 2019
%F A292627 T(n,k) =  (1/Pi) * Integral_{x = -1..1} (k - 2 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (k + 2 - 4*x^2)^n/sqrt(1 - x^2) dx. - _Peter Bala_, Jan 27 2020
%F A292627 A(n,k) = (1/4)^n * Sum_{j=0..n} (k-2)^j * (k+2)^(n-j) * binomial(2*j,j) * binomial(2*(n-j),n-j). - _Seiichi Manyama_, Aug 18 2025
%e A292627 E.g.f. of column k: A_k(x) =  1 + k*x/1! + (k^2 + 2)*x^2/2! + (k^3 + 6*k)*x^3/3! + (k^4 + 12*k^2 + 6)*x^4/4! + (k^5 + 20*k^3 + 30*k)*x^5/5! + ...
%e A292627 Square array begins:
%e A292627   1,   1,    1,    1,     1,     1,  ...
%e A292627   0,   1,    2,    3,     4,     5,  ...
%e A292627   2,   3,    6,   11,    18,    27,  ...
%e A292627   0,   7,   20,   45,    88,   155,  ...
%e A292627   6,  19,   70,  195,   454,   931,  ...
%e A292627   0,  51,  252,  873,  2424,  5775,  ...
%t A292627 Table[Function[k, n! SeriesCoefficient[Exp[k x] BesselI[0, 2 x], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%t A292627 Table[Function[k, SeriesCoefficient[1/Sqrt[(1 + 2 x - k x) (1 - 2 x - k x)], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y A292627 Columns k=0..7 give A126869, A002426, A000984, A026375, A081671, A098409, A098410, A104454.
%Y A292627 Rows n=0..2 give A000012, A001477, A059100.
%Y A292627 Main diagonal gives A186925.
%Y A292627 Cf. A292628, A307847.
%K A292627 nonn,tabl
%O A292627 0,6
%A A292627 _Ilya Gutkovskiy_, Sep 20 2017