A055864 -id:A055864 - cp's OEIS frontend

A055865 Second column of triangle A055864.

0, 2, 12, 100, 1080, 14406, 229376, 4251528, 90000000, 2143588810, 56757583872, 1654301902188, 52644347205632, 1816448730468750, 67553994410557440, 2694045224950414864, 114692890480116793344, 5191945444217181018258, 249036800000000000000000, 12617615847934310595791220

Offset: 1

Wolfdieter Lang Jun 20 2000

Paolo Xausa, Table of n, a(n) for n = 1..350
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A055864, A000272, A055858.

A055865[n_] := If[n == 1, 0, n*(n+1)^(n-2)];
Array[A055865, 25] (* Paolo Xausa, Aug 07 2025 *)

a(1) = 0; a(n) = n*(n+1)^(n-2), n >= 2.
E.g.f.: -1/2*(-W(-x)^2+x^2)/x, W(x) principal branch of Lambert's function.
a(n) = A055864(n, 2).

A055070 Third column of triangle A055864.

Original entry on oeis.org

0, 0, 9, 80, 900, 12348, 200704, 3779136, 81000000, 1948717100, 52027785216, 1527047909712, 48884036690944, 1695352148437500, 63331869759897600, 2535571976423919872, 108321063231221415936, 4918685157679434648876

Offset: 1

Wolfdieter Lang Jun 20 2000

Cf. A055864, A000272, A055865.

a(1)=0= a(2); a(n)= n^2*(n+1)^(n-3), n >= 3.
E.g.f. (3*W(-x)^2-2*W(-x)^3-3*x^2-8*x^3)/(12*x), W(x) principal branch of Lambert's function.
a(n)=A055864(n, 3).

A055867 Fourth column of triangle A055864.

Original entry on oeis.org

0, 0, 0, 64, 750, 10584, 175616, 3359232, 72900000, 1771561000, 47692136448, 1409582685888, 45392319784448, 1582328671875000, 59373627899904000, 2386420683693101056, 102303226385042448384, 4659806991485780193672

Offset: 1

Wolfdieter Lang, Jun 20 2000

a(n)=A055864(n, 4). Cf. A000272, A055865, A055070, A055866, A055858.

A055867:=n->n^3*(n+1)^(n-4): 0,0,0,seq(A055867(n), n=4..30); # Wesley Ivan Hurt, Jan 27 2017

a(i)=0 for i=1, 2, 3; a(n) = n^3*(n+1)^(n-4), n >= 4.

E.g.f.: (9*W(-x)^2-14*W(-x)^3+3*W(-x)^4-9*x^2-32*x^3-81*x^4)/(72*x), W(x) principal branch of Lambert's function.

A055868 Fifth column of triangle A055864.

Original entry on oeis.org

0, 0, 0, 0, 625, 9072, 153664, 2985984, 65610000, 1610510000, 43717791744, 1301153248512, 42150011228416, 1476840093750000, 55662776156160000, 2246042996417036288, 96619713808095645696, 4414553991933897025584

Offset: 1

Wolfdieter Lang Jun 20 2000

a(n)=A055864(n, 5). Cf. A000272, A055865, A055070, A055867, A055858.

a(i)=0 for i=1..4; a(n)= n^4*(n+1)^(n-5), n >= 5.

E.g.f.: (270*W(-x)^2-740*W(-x)^3+345*W(-x)^4-36*W(-x)^5-270*x^2-1280*x^3-3645*x^4-9216*x^5)/(4320*x), W(x) principal branch of Lambert's function.

A055858 Coefficient triangle for certain polynomials.

Original entry on oeis.org

1, 1, 2, 4, 9, 6, 27, 64, 48, 36, 256, 625, 500, 400, 320, 3125, 7776, 6480, 5400, 4500, 3750, 46656, 117649, 100842, 86436, 74088, 63504, 54432, 823543, 2097152, 1835008, 1605632, 1404928, 1229312, 1075648, 941192, 16777216, 43046721

Offset: 0

Wolfdieter Lang, Jun 20 2000

The coefficients of the partner polynomials are found in triangle A055864.

			{1}; {1,2}; {4,9,6}; {27,64,48,36}; ...
Fourth row polynomial (n=3): p(3,x) = 27 + 64*x + 48*x^2 + 36*x^3.

Column sequences are A000312(n), n >= 1, A055860 (A000169), A055861 (A053506), A055862-3 for m=0..4, row sums: A045531(n+1)= |A039621(n+1, 2)|, n >= 0.

a[n_, m_] /; n < m = 0; a[0, 0] = 1; a[n_, 0] := n^n; a[n_, m_] := n^(m-1)*(n+1)^(n-m+1); Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2013 *)

a(n, m)=0 if n < m; a(0, 0)=1, a(n, 0) = n^n, n >= 1, a(n, m) = n^(m-1)*(n+1)^(n-m+1), n >= m >= 1;

E.g.f. for column m: A(m, x); A(0, x) = 1/(1+W(-x)); A(1, x) = -1 - (d/dx)W(-x) = -1-W(-x)/((1+W(-x))*x); A(2, x) = A(1, x)-int(A(1, x), x)/x-(1/x+x); recursion: A(m, x) = A(m-1, x)-int(A(m-1, x), x)/x-((m-1)^(m-1))*(x^(m-1))/(m-1)!, m >= 3; W(x) principal branch of Lambert's function.

A055869 a(n) = (n+1)^n - n^n.

Original entry on oeis.org

1, 5, 37, 369, 4651, 70993, 1273609, 26269505, 612579511, 15937424601, 457696700077, 14381984674225, 490839666661891, 18080919199832609, 715027614225987601, 30214447801957316865, 1358671297852359767791, 64780942222614703957417, 3264460344339686410876021

Offset: 1

Wolfdieter Lang, Jun 20 2000

Number of functions f:[n]->[n+1] such that some x in [n] maps to n+1.

Number of switching generators for a power polyadic n-context ({1..k}, ..., {1..k}, <>) with n=k [Theorems 5 and 6, page 81, in Ignatov]. - Dmitry I. Ignatov, Nov 23 2022

G. C. Greubel, Table of n, a(n) for n = 1..385
D. I. Ignatov, On closure operators related to maximal tricliques in tripartite hypergraphs, Discrete Appl. Math., 249 (2018), 74-84.
D. I. Ignatov, Supporting iPython code for enumeration of switching generators of power polyadic n-contexts for n=k<6, GitHub repository.

Row sums of triangle A055864.

Cf. A055864, A055858, A045531.

[(n+1)^n - n^n: n in [1..40]]; // Vincenzo Librandi, Jan 11 2015

Table[(n+1)^n-n^n,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 20 2009 *)

vector(20, n, (n+1)^n - n^n) \\ Michel Marcus, Jan 10 2015

E.g.f.: W(-x)*(x-1)/((1+W(-x))*x), W(x) principal branch of Lambert's function.
a(n) = Sum_{m=1..n} A055864(n, m).
a(n) = Sum_{i=0..n-1} n^i*C(n, i). - Olivier Gérard, Jun 26 2001
With interpolated zeros, ceiling(n/2)^floor(n/2) - floor(n/2)^floor(n/2). - Paul Barry, Jul 13 2005
a(n) = Sum_{k=1..n} (-1)^(n-k)*k!*Stirling2(n,k)*binomial(n+k-1,n). - Vladimir Kruchinin, Sep 20 2015

More terms from Vincenzo Librandi, Jan 11 2015

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A055865 Second column of triangle A055864.

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A055070 Third column of triangle A055864.

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A055867 Fourth column of triangle A055864.

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A055868 Fifth column of triangle A055864.

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A055858 Coefficient triangle for certain polynomials.

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A055869 a(n) = (n+1)^n - n^n.

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