A078751 Triangle read by rows: T(m,k) = normalized partial derivative of (t,z) -> exp(t*g(z)) at (0,0), where 2*g(z) = 1 + exp(-2*z*g(z)).

2, 4, 8, 24, 48, 48, 224, 480, 576, 384, 2880, 6400, 8640, 7680, 3840, 47232, 107520, 155520, 161280, 115200, 46080, 942592, 2182656, 3306240, 3763200, 3225600, 1935360, 645120, 22171648, 51996672, 81414144, 98703360, 94617600, 69672960

Offset: 0

Author

Carmen Chicone (carmen(AT)chicone.math.missouri.edu), Dec 22 2002

Keywords

• nonn
• tabl
• easy

Comments

Let g(z) = 1/2 + W(z/e^z) / (2*z), where W is Lambert's W-function; g satisfies 2 * g(z) = 1 + exp(-2 * z *(z)). Let c(m,n) be the coefficient of z^m in the Maclaurin series for g(z)^n; equivalently c(m,n) is 1/m! times the mixed partial derivative (d^(m+n) f(t,z)) / (dz^m dt^n), where f(t,z) = exp(t*g(z)). For 0

Examples

			Triangle begins:
  2;
  4, 8;
  24, 48, 48;
  224, 480, 576, 384;
  ...
		

Crossrefs

First column of triangular array (T(m, 1) for m>=1) is A038049.

Programs

Mathematica

(* ccctri lists first numrows rows of triangular array. *)
ccctri[numrows_] := (s[j_] := Sum[Binomial[j, i] i^(j-1), {i, 1, j}]; r[j_] := 1/2 (-1)^j 1/(j+1)! s[j+1]; c[m_, k_] := 1/m Sum[((k+1) j-m)c[m-j, k]r[j], {j, 1, m}]; c[0, k_] := 1; ss[m_, k_] := 2^k m! (-1)^(m-k) c[m-k, k]; Flatten[Table[Table[ss[k, j], {j, 1, k}], {k, 1, numrows}]])
(* ccccol lists maxrow elements of column colnum. *)
ccccol[colnum_, maxrow_] := (s[j_] := Sum[Binomial[j, i] i^(j-1), {i, 1, j}]; r[j_] := 1/2 (-1)^j 1/(j+1)! s[j+1]; c[m_, k_] := 1/m Sum[((k+1) j-m)c[m-j, k]r[j], {j, 1, m}]; c[0, k_] := 1; ss[m_, k_] := 2^k m! (-1)^(m-k) c[m-k, k]; Table[ss[m, colnum], {m, colnum, maxrow}])

Formula

T(n, k) = 2^k * n! * (-1)^(n-k) * c(n-k,k) where c(n, k) = (1/n) * Sum_{j=1..n} (((k+1)*j-n) * c(n-j, k) * c(j, 1)), where c(0, k)=1 and c(j, 1) = (1/2) * (-1)^j * (1/(j+1)!) * Sum_{i=1..j+1} binomial(j+1, i) * i^j.

Extensions

Edited by Dean Hickerson, Dec 30 2002

Revised by Sean A. Irvine, Jul 14 2025

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