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A108290
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Triangle, read by rows, such that row n equals the inverse binomial transform of row n of table A060543, where A060543(n,k) = C(n+n*k+k, n*k+k).
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4
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1, 1, 2, 1, 9, 9, 1, 34, 96, 64, 1, 125, 750, 1250, 625, 1, 461, 5265, 16470, 19440, 7776, 1, 1715, 35329, 184877, 386561, 352947, 117649, 1, 6434, 232288, 1913408, 6307840, 9863168, 7340032, 2097152, 1, 24309, 1513656, 18921924, 92365758, 220016574
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OFFSET
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0,3
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COMMENTS
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The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (1+n*x)*(2+n*x)*...*(n-1+n*x)/(n-1)! in the basis made of the binomial(x+i,i). - F. Chapoton, Nov 01 2022
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LINKS
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EXAMPLE
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BINOMIAL[1, 9, 9] = {1, 10, 28, 55, 91, 136, 190, 253, ...}.
BINOMIAL[1, 34, 96, 64] = {1, 35, 165, 455, 969, 1771, 2925, ...}.
BINOMIAL[1, 125, 750, 1250, 625] = {1, 126, 1001, 3876, 10626, ...}.
Triangle begins:
1;
1, 2;
1, 9, 9;
1, 34, 96, 64;
1, 125, 750, 1250, 625;
1, 461, 5265, 16470, 19440, 7776;
1, 1715, 35329, 184877, 386561, 352947, 117649;
1, 6434, 232288, 1913408, 6307840, 9863168, 7340032, 2097152; ...
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PROG
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(PARI) {T(n, k)=local(X=x+x*O(x^k)); polcoeff(sum(j=0, n, binomial(n+n*j+j, n*j+j)*(x/(1+X))^j)/(1+X), k)}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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