A137452 -id:A137452 - cp's OEIS frontend

A362356 a(n) = 5*(n + 5)^(n-1).

1, 5, 35, 320, 3645, 50000, 805255, 14929920, 313742585, 7378945280, 192216796875, 5497558138880, 171359481538165, 5784156907130880, 210264917311285295, 8192000000000000000, 340611592914758411505, 15056807481695325716480, 705250197803314844630515

Offset: 0

Wolfdieter Lang, Apr 24 2023

This gives the fifth exponential (also called binomial) convolution of {A000272(n+1)} = {A232006(n+1, 1)}, for n >= 0, with e.g.f. (LambertW(-x),(-x)) (LambertW is the principal branch of the Lambert W-function).

This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = 5.

Eric Weisstein's World of Mathematics, Lambert W-function
Wikipedia, Lambert W function

Column k=5 of A232006 (without leading zeros).

Cf. A000272, A137452.

a(n) = Sum_{k=0..n} |A137452(n, k)|*5^k = Sum_{k=0..n} binomial(n-1, k-1)*n^(n-k)*5^k, with the n = 0 term equal to 1 (not 0).
E.g.f.: (LambertW(-x)/(-x))^5.
From Seiichi Manyama, Jun 19 2024: (Start)
E.g.f. A(x) satisfies:
(1) A(x) = exp(5*x*A(x)^(1/5)).
(2) A(x) = 1/A(-x*A(x)^(2/5)). (End)

A367254 a(n) = binomial(2n - 1, n - 1)(2*n)^n.

Original entry on oeis.org

1, 2, 48, 2160, 143360, 12600000, 1379524608, 180889572864, 27638114549760, 4822114348846080, 945950720000000000, 206098414000597966848, 49378358320648503164928, 12902739286521391316172800, 3651796443284936332620595200, 1112883434275320000000000000000

Offset: 0

Peter Luschny, Nov 11 2023

nonn

Cf. A137452.

a := n -> binomial(2*n - 1, n - 1)*(2*n)^n;
seq(a(n), n = 0..15);

A367254[n_]:=If[n==0,1,Binomial[2n-1,n-1](2n)^n];
Array[A367254,20,0] (* Paolo Xausa, Dec 01 2023 *)

a(n) = abs(A137452(2*n, n)).

A367255 a(n) = (n + 1)^(n - 2)(3n + 1).

Original entry on oeis.org

1, 2, 7, 40, 325, 3456, 45619, 720896, 13286025, 280000000, 6645125311, 175432531968, 5100764198413, 161982606786560, 5579092529296875, 207165582859042816, 8250513501410645521, 350825312056827838464, 15864277746219164222455, 760217600000000000000000

Offset: 0

Peter Luschny, Nov 11 2023

nonn

Cf. A137452.

a := n -> (n + 1)^(n - 2)*(3*n + 1): seq(a(n), n = 0..19);

A367255[n_]:=(n+1)^(n-2)(3n+1);
Array[A367255,25,0] (* Paolo Xausa, Dec 01 2023 *)

a(n) = Sum_{k=0..n} binomial(n - 1, k - 1) * n^(n - k) * (k + 1). (See A137452.)

cp's OEIS Frontend

A362356 a(n) = 5*(n + 5)^(n-1).

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Formula

A367254 a(n) = binomial(2n - 1, n - 1)(2*n)^n.

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Maple

Mathematica

Formula

A367255 a(n) = (n + 1)^(n - 2)(3n + 1).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

cp's OEIS Frontend

A362356 a(n) = 5*(n + 5)^(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A367254 a(n) = binomial(2*n - 1, n - 1)*(2*n)^n.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A367255 a(n) = (n + 1)^(n - 2)*(3*n + 1).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A367254 a(n) = binomial(2n - 1, n - 1)(2*n)^n.

A367255 a(n) = (n + 1)^(n - 2)(3n + 1).