A360660 -id:A360660 - cp's OEIS frontend

A154323 Central coefficients of number triangle A113582.

1, 2, 10, 37, 101, 226, 442, 785, 1297, 2026, 3026, 4357, 6085, 8282, 11026, 14401, 18497, 23410, 29242, 36101, 44101, 53362, 64010, 76177, 90001, 105626, 123202, 142885, 164837, 189226, 216226, 246017, 278785, 314722, 354026, 396901, 443557, 494210, 549082, 608401, 672401, 741322, 815410, 894917, 980101

Offset: 0

Paul Barry, Jan 07 2009

a(n) equals n!^3 times the determinant of the n X n matrix whose (i,j)-entry is KroneckerDelta[i, j] (((i^3 + 1)/(i^3)) - 1) + 1. - John M. Campbell, May 20 2011
Let b(0)=b(1)=1; b(n)=max(b(n-1)+(n-1)^3, b(n-2)+(n-2)^3); then a(n)=b(n+1). - Yalcin Aktar, Jul 28 2011
a(n-1) is the number of sets of n words of length n over binary alphabet where the first letter occurs n times. a(2) = 10: {aab,abb,bbb}, {aab,bab,bbb}, {aab,bba,bbb}, {aba,abb,bbb}, {aba,bab,bbb}, {aba,bba,bbb}, {abb,baa,bbb}, {abb,bab,bba}, {baa,bab,bbb}, {baa,bba,bbb}. - Alois P. Heinz, Feb 16 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537, A113582, A360660.

Main diagonal of A360693.

[(n^4 + 2*n^3 + n^2 + 4)/4: n in [0..40]]; // Vincenzo Librandi, Feb 13 2015

s = 1; lst = {s}; Do[s += n^3; AppendTo[lst, s], {n, 1, 42, 1}]; lst (* Zerinvary Lajos, Jul 12 2009 *)
Table[n!^3*Det[Array[KroneckerDelta[#1,#2](((#1^3+1)/(#1^3))-1)+1&,{n,n}]],{n,1,30}] (* John M. Campbell, May 20 2011 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 2, 10, 37, 101}, 25] (* or *) Table[(n^4 + 2*n^3 + n^2 + 4)/4, {n,0,25}] (* G. C. Greubel, Sep 11 2016 *)

a(n) = (n^4 + 2*n^3 + n^2 + 4)/4.
G.f.: (1 - 3*x + 10*x^2 - 3*x^3 + x^4)/(1-x)^5.
a(n) = 1 + C(n+1,2)^2 = 1 + A000537(n).
From G. C. Greubel, Sep 11 2016: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
E.g.f.: (1/4)*(4 + 4*x + 14*x^2 + 8*x^3 + x^4)*exp(x). (End)
a(n) = a(n-1)+n^3. - Charles U. Lonappan, Jun 09 2021

A220886 Irregular triangular array read by rows: T(n,k) is the number of inequivalent n X n {0,1} matrices modulo permutation of the rows, containing exactly k 1's; n>=0, 0<=k<=n^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 9, 20, 27, 27, 20, 9, 3, 1, 1, 4, 16, 48, 133, 272, 468, 636, 720, 636, 468, 272, 133, 48, 16, 4, 1, 1, 5, 25, 95, 330, 1027, 2780, 6550, 13375, 23700, 36403, 48405, 55800, 55800, 48405, 36403, 23700, 13375, 6550, 2780, 1027, 330, 95, 25, 5, 1

Offset: 0

Geoffrey Critzer, Feb 20 2013

In other words, two matrices are considered equivalent if one can be obtained from the other by some sequence of interchanges of the rows.

			T(2,2) = 4 because we have: {{0,0},{1,1}}; {{0,1},{1,0}}; {{0,1},{0,1}}; {{1,0},{1,0}} (where the first two matrices were arbitrarily selected as class representatives).
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2,  4,  2,   1;
  1, 3,  9, 20,  27,  27,  20,   9,   3,   1;
  1, 4, 16, 48, 133, 272, 468, 636, 720, 636, 468, 272, 133, 48, 16, 4, 1;
  ...

Alois P. Heinz, Rows n = 0..32, flattened

Row sums are A060690.
Columns k=0-3 give: A000012, A000027, A000290 (n>=2), A203552 (n>=3).
Main diagonal gives A360660.
Cf. A360693.

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$3)):
seq(T(n), n=0..5);  # Alois P. Heinz, Feb 15 2023

nn=100;Table[CoefficientList[Series[CycleIndex[SymmetricGroup[n],s]/.Table[s[i]->(1+x^i)^n,{i,1,n}],{x,0,nn}],x],{n,0,5}]//Grid
(* Second program: *)
g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i] + k - 1, k], {k, 0, j}]]]];
T[n_] := CoefficientList[g[n, n, n], x];
Table[T[n], {n, 0, 5}] // Flatten (* Jean-François Alcover, May 28 2023, after Alois P. Heinz *)

A383073 a(n) = [x^n] Product_{k=1..n} (1 + x^k)^binomial(n,k).

Original entry on oeis.org

1, 1, 2, 11, 69, 552, 5133, 53804, 626440, 7979043, 110074741, 1631532542, 25813521836, 433619035254, 7698641650937, 143908414079881, 2822753485000135, 57930283521990154, 1240695879627856673, 27666701629865989070, 641049490249340264699

Offset: 0

Vaclav Kotesovec, Apr 15 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A360660.

Cf. A026007, A102866, A344130, A369578.

Table[SeriesCoefficient[Product[(1 + x^k)^Binomial[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 20}]

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A154323 Central coefficients of number triangle A113582.

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A220886 Irregular triangular array read by rows: T(n,k) is the number of inequivalent n X n {0,1} matrices modulo permutation of the rows, containing exactly k 1's; n>=0, 0<=k<=n^2.

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A383073 a(n) = [x^n] Product_{k=1..n} (1 + x^k)^binomial(n,k).

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