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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.

%I A375466 #11 Mar 30 2025 06:32:13
%S A375466 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,
%T A375466 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,
%U A375466 54,6,1,0,1,10,1,128,14,1,750,96,9,1,24,0
%N 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.
%F A375466 T(n, m, k) = [x^k] n! * m^n * hypergeom([-n], [-n], x/m), for n > 0.
%e A375466 Sequence of polynomials P(n, m) for n = 0, 1, 2, ...:
%e A375466   [0]   1;
%e A375466   [1]   1*m   +         x;
%e A375466   [2]   2*m^2 +     2*m*x +         x^2;
%e A375466   [3]   6*m^3 +   6*m^2*x +     3*m*x^2 +         x^3;
%e A375466   [4]  24*m^4 +  24*m^3*x +  12*m^2*x^2 +     4*m*x^3 +        x^4;
%e A375466   [5] 120*m^5 + 120*m^4*x +  60*m^3*x^2 +  20*m^2*x^3 +    5*m*x^4 +     x^5;
%e A375466   [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;
%e A375466   ...
%e A375466 Array of the coefficients of the polynomials for m = 0, 1, 2, ...:
%e A375466   [0] 1, 0, 1,  0,  0, 1,    0,   0,  0, 1,     0,    0,   0,  0, 1, ...  A023531
%e A375466   [1] 1, 1, 1,  2,  2, 1,    6,   6,  3, 1,    24,   24,  12,  4, 1, ...  A094587
%e A375466   [2] 1, 2, 1,  8,  4, 1,   48,  24,  6, 1,   384,  192,  48,  8, 1, ...
%e A375466   [3] 1, 3, 1, 18,  6, 1,  162,  54,  9, 1,  1944,  648, 108, 12, 1, ...
%e A375466   [4] 1, 4, 1, 32,  8, 1,  384,  96, 12, 1,  6144, 1536, 192, 16, 1, ...
%e A375466   [5] 1, 5, 1, 50, 10, 1,  750, 150, 15, 1, 15000, 3000, 300, 20, 1, ...
%e A375466   [6] 1, 6, 1, 72, 12, 1, 1296, 216, 18, 1, 31104, 5184, 432, 24, 1, ...
%e A375466   ...
%e A375466 Seen as triangle:
%e A375466   1;
%e A375466   1, 0;
%e A375466   1, 1, 1;
%e A375466   1, 2, 1,  0;
%e A375466   1, 3, 1,  2,  0;
%e A375466   1, 4, 1,  8,  2, 1;
%e A375466   1, 5, 1, 18,  4, 1,   0;
%e A375466   1, 6, 1, 32,  6, 1,   6,  0;
%e A375466   1, 7, 1, 50,  8, 1,  48,  6, 0;
%e A375466   1, 8, 1, 72, 10, 1, 162, 24, 3, 1;
%e A375466   1, 9, 1, 98, 12, 1, 384, 54, 6, 1,  0;
%p A375466 # Computes the polynomials depending on the parameter m.
%p A375466 P := (n, m) -> ifelse(m = 0, x^n, n! * m^n * hypergeom([-n], [-n], x/m)):
%p A375466 seq(print(simplify(P(n, m))), n = 0..5);
%p A375466 # Computes the array of coefficients:
%p A375466 P := (n, k, m) -> (n!/k!) * m^(n-k) * x^k:
%p A375466 Arow := (m, len) -> local n, k;
%p A375466 seq(seq(coeff(P(n, k, m), x, k), k = 0..n), n = 0..len):
%p A375466 seq(lprint(Arow(n, 4)), n = 0..6);
%Y A375466 Cf. A094587, A023531.
%K A375466 nonn,tabl
%O A375466 0,8
%A A375466 _Peter Luschny_, Aug 17 2024