A038049 -id:A038049 - cp's OEIS frontend

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

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

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

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

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

(* 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}])

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.

Edited by Dean Hickerson, Dec 30 2002

Revised by Sean A. Irvine, Jul 14 2025

A210627 Constants r_n arising in study of polynomials of least deviation from zero in several variables.

Original entry on oeis.org

72, 896, 14400, 283392, 6598144, 177373184, 5406289920, 184223744000, 6939874934784, 286375842938880, 12846564299505664, 622448445155704832, 32395710363284275200, 1802446793652649852928, 106760825994912064339968, 6707088257932303257305088, 445456559121345605093294080, 31185504805980142781333504000

Offset: 3

N. J. A. Sloane, Mar 25 2012

nonn

Yuan Xu, On polynomials of least deviation from zero in several variables, J. Experimental Math. 13 (2004), 103-112.

For n>3, r_n = n*Sum_{k=4..n} k^(n-3) binomial(n,k) [(-1)^k(9k^2-32k+24)+k^2].

Conjecture: a(n) = n*A038049(n). - R. J. Mathar, Mar 27 2012

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A038054 Number of labeled trees with 2-colored leaves.

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

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A210627 Constants r_n arising in study of polynomials of least deviation from zero in several variables.

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Author

Keywords

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Formula

cp's OEIS Frontend

A038054 Number of labeled trees with 2-colored leaves.

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Author

Keywords

Links

Crossrefs

Formula

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)).

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Author

Keywords

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Examples

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Programs

Mathematica

Formula

Extensions

A210627 Constants r_n arising in study of polynomials of least deviation from zero in several variables.

Views

Author

Keywords

Links

Formula

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