A375466 Array read by ascending antidiagonals of triangles read by rows: the coefficients of the polynomials n! * m^(n-k) * x^k * A094587(n, k), for m >= 0.
1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 3, 1, 2, 0, 1, 4, 1, 8, 2, 1, 1, 5, 1, 18, 4, 1, 0, 1, 6, 1, 32, 6, 1, 6, 0, 1, 7, 1, 50, 8, 1, 48, 6, 0, 1, 8, 1, 72, 10, 1, 162, 24, 3, 1, 1, 9, 1, 98, 12, 1, 384, 54, 6, 1, 0, 1, 10, 1, 128, 14, 1, 750, 96, 9, 1, 24, 0
Offset: 0
Examples
Sequence of polynomials P(n, m) for n = 0, 1, 2, ...: [0] 1; [1] 1*m + x; [2] 2*m^2 + 2*m*x + x^2; [3] 6*m^3 + 6*m^2*x + 3*m*x^2 + x^3; [4] 24*m^4 + 24*m^3*x + 12*m^2*x^2 + 4*m*x^3 + x^4; [5] 120*m^5 + 120*m^4*x + 60*m^3*x^2 + 20*m^2*x^3 + 5*m*x^4 + x^5; [6] 720*m^6 + 720*m^5*x + 360*m^4*x^2 + 120*m^3*x^3 + 30*m^2*x^4 + 6*m*x^5 + x^6; ... Array of the coefficients of the polynomials for m = 0, 1, 2, ...: [0] 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, ... A023531 [1] 1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 24, 24, 12, 4, 1, ... A094587 [2] 1, 2, 1, 8, 4, 1, 48, 24, 6, 1, 384, 192, 48, 8, 1, ... [3] 1, 3, 1, 18, 6, 1, 162, 54, 9, 1, 1944, 648, 108, 12, 1, ... [4] 1, 4, 1, 32, 8, 1, 384, 96, 12, 1, 6144, 1536, 192, 16, 1, ... [5] 1, 5, 1, 50, 10, 1, 750, 150, 15, 1, 15000, 3000, 300, 20, 1, ... [6] 1, 6, 1, 72, 12, 1, 1296, 216, 18, 1, 31104, 5184, 432, 24, 1, ... ... Seen as triangle: 1; 1, 0; 1, 1, 1; 1, 2, 1, 0; 1, 3, 1, 2, 0; 1, 4, 1, 8, 2, 1; 1, 5, 1, 18, 4, 1, 0; 1, 6, 1, 32, 6, 1, 6, 0; 1, 7, 1, 50, 8, 1, 48, 6, 0; 1, 8, 1, 72, 10, 1, 162, 24, 3, 1; 1, 9, 1, 98, 12, 1, 384, 54, 6, 1, 0;
Programs
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Maple
# Computes the polynomials depending on the parameter m. P := (n, m) -> ifelse(m = 0, x^n, n! * m^n * hypergeom([-n], [-n], x/m)): seq(print(simplify(P(n, m))), n = 0..5); # Computes the array of coefficients: P := (n, k, m) -> (n!/k!) * m^(n-k) * x^k: Arow := (m, len) -> local n, k; seq(seq(coeff(P(n, k, m), x, k), k = 0..n), n = 0..len): seq(lprint(Arow(n, 4)), n = 0..6);
Formula
T(n, m, k) = [x^k] n! * m^n * hypergeom([-n], [-n], x/m), for n > 0.