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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).
12
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, 27, 88, 195, 252, 141, 0, 1, 7, 38, 155, 454, 873, 924, 393, 70, 1, 8, 51, 252, 931, 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
OFFSET
0,6
COMMENTS
A(n,k) is the k-th binomial transform of A126869 evaluated at n.
LINKS
N. J. A. Sloane, Transforms
FORMULA
O.g.f. of column k: 1/sqrt( (1 - (k-2)*x)*(1 - (k+2)*x) ).
E.g.f. of column k: exp(k*x)*BesselI(0,2*x).
From Seiichi Manyama, May 01 2019: (Start)
A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + x^2)^n.
A(n,k) = Sum_{j=0..n} (k-2)^(n-j) * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..n} (k+2)^(n-j) * (-1)^j * binomial(n,j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - (k^2-4) * (n-1) * A(n-2,k). (End)
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
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
EXAMPLE
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! + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
2, 3, 6, 11, 18, 27, ...
0, 7, 20, 45, 88, 155, ...
6, 19, 70, 195, 454, 931, ...
0, 51, 252, 873, 2424, 5775, ...
MATHEMATICA
Table[Function[k, n! SeriesCoefficient[Exp[k x] BesselI[0, 2 x], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
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
CROSSREFS
Rows n=0..2 give A000012, A001477, A059100.
Main diagonal gives A186925.
Sequence in context: A092872 A364880 A141455 * A113125 A088239 A130070
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Sep 20 2017
STATUS
approved