A060063 -id:A060063 - cp's OEIS frontend

A091742 Second column (m=1) of triangle A060063.

1, 26, 775, 32516, 1894429, 148008446, 15005570563, 1920091871816, 303034557325177, 57867367557774626, 13156672434190398271, 3513034745241511676876, 1088776741031904909773845

Offset: 1

Wolfdieter Lang, Feb 13 2004

a(n)= A060063(n, 1), n>=1.

A091743 Third column (m=2) of triangle A060063 divided by 9.

Original entry on oeis.org

1, 131, 12702, 1331426, 163594663, 23979017805, 4187448766444, 864808560876788, 209300424292684605, 58811116750666789895, 19017126570296402361706, 7018937700334256743262646

Offset: 2

Wolfdieter Lang, Feb 13 2004

a(n)= A060063(n, 2)/9, n>=2.

A091744 Fourth column (m=3) of triangle A060063 divided by 9.

Original entry on oeis.org

25, 9684, 2263474, 490886452, 111812888083, 27999019489624, 7844432790401028, 2473024180429410040, 877529810239038245325, 349669921564112871530060, 155914953071171948587819830, 77481281483613804213513143564

Offset: 3

Wolfdieter Lang, Feb 13 2004

A091745 Alternating row sums of triangle A060063.

Original entry on oeis.org

1, 0, -12, 240, 7056, -1472640, 60785472, 24634471680, -6666732039936, -266945607045120, 818181307311879168, -225753930347735961600, -114700401016766429097984, 133755245359458345703342080

Offset: 0

Wolfdieter Lang, Feb 13 2004

a(n)=sum(A060063(n, m)*(-1)^m, m=0..n), n>=0.

A052502 Number of permutations sigma of [3n] without fixed points such that sigma^3 = Id.

Original entry on oeis.org

1, 2, 40, 2240, 246400, 44844800, 12197785600, 4635158528000, 2345390215168000, 1524503639859200000, 1237896955565670400000, 1227993779921145036800000, 1461312598106162593792000000, 2054605512937264606871552000000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

For n >= 1 a(n) is the size of the conjugacy class in the symmetric group S_(3n) consisting of permutations whose cycle decomposition is a product of n disjoint 3-cycles.

F. W. J. Olver, Asymptotics and special functions, Academic Press, 1974, pages 336-344.

G. C. Greubel, Table of n, a(n) for n = 0..210
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 27
F. W. J. Olver et al., NIST Digital Library of Mathematical Functions, eq. 9.10.17.

Cf. A000142. Row sums of triangle A060063.
First column of array A091752 (also negative of second column).
Equals row sums of A157702. - Johannes W. Meijer, Mar 07 2009
Karol A. Penson suggested that the row sums of A060063 coincide with this entry.
Cf. A001147, A052504, A060706, A261317, A261381.
Trisection of column k=3 of A261430.

List([0..20], n-> Factorial(3*n)/(3^n*Factorial(n))) # G. C. Greubel, May 14 2019

[Factorial(3*n)/(3^n*Factorial(n)): n in [0..20]]; // G. C. Greubel, May 14 2019

spec := [S,{S=Set(Union(Cycle(Z,card=3)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[(3*n)!/(3^n*n!), {n, 0, 20}] (* G. C. Greubel, May 14 2019 *)

{a(n) = (3*n)!/(3^n*n!)}; \\ G. C. Greubel, May 14 2019

[factorial(3*n)/(3^n*factorial(n)) for n in (0..20)] # G. C. Greubel, May 14 2019

From Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001: (Start)
a(n) = (3*n)!/(3^n * n!).
a(n) ~ sqrt(3) * 9^n * (n/e)^(2n). (End)
E.g.f.: (every third coefficient of) exp(x^3/3).
G.f.: hypergeometric3F0([1/3, 2/3, 1], [], 9*x).
D-finite with recurrence a(n) = (3*n-1)*(3*n-2)*a(n-1) for n >= 1, with a(0) = 1.
Write the generating function for this sequence in the form A(x) = Sum_{n >= 0} a(n)* x^(2*n+1)/(2*n+1)!. The g.f. A(x) satisfies A'(x)*( 1 - A(x)^2) = 1. Robert Israel remarks that consequently A(x) is a root of z^3 - 3*z + 3*x with A(0) = 0. Cf. A001147, A052504 and A060706. - Peter Bala, Jan 02 2015
From Peter Bala, Feb 27 2024: (Start)
u(n) := a(n+1) satisfies the second-order recurrence u(n) = 18*n*u(n-1) + (3*n - 1)^2*(3*n - 2)^2*u(n-2) with u(0) = 2 and u(1) = 40.
A second solution to the recurrence is given by v(n) := u(n)*Sum_{k = 0..n} (-1)^k/((3*k + 1)*(3*k + 2)) with v(0) = 1 and v(1) = 18.
This leads to the continued fraction expansion (2/3)*log(2) = Sum_{k = 0..n} (-1)^k/((3*k + 1)*(3*k + 2)) = Limit_{n -> oo} v(n)/u(n) = 1/(2 + (1*2)^2/(18 + (4*5)^2/(2*18 + (7*8)^2/(3*18 + (10*11)^2/(4*18 +  ... ))))). (End)
From Gabriel B. Apolinario, Jul 30 2024: (Start)
a(n) = 3 * Integral_{t=0..oo} Ai(t)*t^(3*n) dt, where Ai(t) is the Airy function.
a(n) = Integral_{t=-oo..oo} Ai(t)*t^(3*n) dt. (End)

Edited by Wolfdieter Lang, Feb 13 2004

Title improved by Geoffrey Critzer, Aug 14 2015

A060058 Triangle of numbers related to A000330 (sum of squares) and A000364 (Euler numbers).

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 1, 14, 61, 61, 1, 30, 331, 1385, 1385, 1, 55, 1211, 12284, 50521, 50521, 1, 91, 3486, 68060, 663061, 2702765, 2702765, 1, 140, 8526, 281210, 5162421, 49164554, 199360981, 199360981, 1, 204, 18522, 948002, 28862471, 510964090, 4798037791, 19391512145, 19391512145

Offset: 0

Wolfdieter Lang, Mar 16 2001

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,   1;
  [2] 1,   5,    5;
  [3] 1,  14,   61,     61;
  [4] 1,  30,  331,   1385,    1385;
  [5] 1,  55, 1211,  12284,   50521,    50521;
  [6] 1,  91, 3486,  68060,  663061,  2702765,   2702765;
  [7] 1, 140, 8526, 281210, 5162421, 49164554, 199360981, 199360981;
  ...

Wolfdieter Lang, First 9 rows.

Cf. A060059 (row sums), A000364 (main diagonal Euler numbers).
Columns: A000012 (powers of 1), A000330 (sum of squares), A060060-2 for m=0,...,4.
See triangle A060074.

T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1) else (n - k + 1)^2 * T(n, k - 1) + T(n - 1, k) fi fi end:
seq(print(seq(T(n, k), k=0..n)), n=0..7);  # Peter Luschny, Sep 30 2023

a[, -1] = 0; a[0, 0] = 1; a[n, m_] /; n < m = 0; a[n_, m_] := a[n, m] = a[n-1, m] + (n+1-m)^2*a[n, m-1]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2013 *)

a(n, m) = a(n-1, m) + ((n+1-m)^2)*a(n, m-1), a(n, -1) := 0, a(0, 0) = 1, a(n, m) = 0 if n < m.
a(n, m) = ay(n-m+1, m) if n >= m >= 0, with the rectangular array ay(n, m) := Sum_{j=1..n} (j^2)*ay(j+1, m-1), n >= 0, m >= 1; input: ay(n, 0)=1 (iterated sums of squares).
G.f. for m-th column: 1/(1-x) for m=0, (x^m)*(Sum_{k=0..m} A060063(m, k)*x^k)/(1-x)^(3*m+1), m >= 1.
Recursion for g.f.s for m-th column: (1-x)*G(m, x) = x*G''(m-1, x) - G'(m-1, x) + G(m-1, x)/x, m >= 2; G(1, x) = x*(1+x)/(1-x)^4; the apostrophe denotes differentiation w.r.t. x. G(0, x) = 1/(1-x). - Wolfdieter Lang, Feb 13 2004

A060061 Fourth column of triangle A060058.

Original entry on oeis.org

61, 1385, 12284, 68060, 281210, 948002, 2749340, 7097948, 16700255, 36419955, 74551048, 144631240, 267951892, 476948260, 819683560, 1365672424, 2213323585, 3499318141, 5410278500, 8197124100

Offset: 0

Wolfdieter Lang, Mar 16 2001

Indranil Ghosh, Table of n, a(n) for n = 0..5000

Cf. A000330, A060060.

Table[Binomial[n+6,6]*(280*n^3+2436*n^2+5906n+3843)/63,{n,0,19}] (* Indranil Ghosh, Feb 21 2017 *)

import math
def C(n, r):
    f=math.factorial
    return f(n)//f(r)//f(n-r)
def A060061(n):
    return (C(n+6, 6)*(280*n**3+2436*n**2+5906*n+3843))//63 # Indranil Ghosh, Feb 21 2017

a(n) = Sum_{j3=1..n+1} j3^2*Sum_{j2=1..j3+1} j2^2*Sum_{j1=1..j2+1} j1^2.
a(n) = A060058(n+3, 3) = binomial(n+6, 6)*(280*n^3+2436*n^2+5906*n+3843)/(7*9).
G.f.: (61+775*x+1179*x^2+225*x^3)/(1-x)^10 = p(3, x)/(1-x)^(3*3+1) with p(3, x)=sum(A060063(3, m)*x^m, m=0..3).

A060060 Third column of triangle A060058.

Original entry on oeis.org

5, 61, 331, 1211, 3486, 8526, 18522, 36762, 67947, 118547, 197197, 315133, 486668, 729708, 1066308, 1523268, 2132769, 2933049, 3969119, 5293519, 6967114, 9059930, 11652030, 14834430, 18710055, 23394735

Offset: 0

Wolfdieter Lang, Mar 16 2001

			a(3) = binomial(7,4) * (20 * 3^2 + 88*3 +75) / 15 = (35 * 519)/15 = 1211. - _Indranil Ghosh_, Feb 21 2017

Indranil Ghosh, Table of n, a(n) for n = 0..5000

Table[(Binomial[n+4,4]*(20*n^2+88*n+75)/15),{n,0,25}] (* Indranil Ghosh, Feb 21 2017 *)

import math
def C(n, r):
    f=math.factorial
    return f(n)//f(r)//f(n-r)
def A060060(n):
    return C(n+4, 4)*(20*n**2+88*n+75)//15 # Indranil Ghosh, Feb 21 2017

a(n) = A060058(n+2, 2) = binomial(n+4, 4)*(20*n^2+88*n+75)/(3*5).

G.f.: (5+26*x+9*x^2)/(1-x)^7 = p(2, x)/(1-x)^(2*3+1). p(2, x)=sum(A060063(2, m)*x^m, m=0..2).

A060062 Fifth column of triangle A060058.

Original entry on oeis.org

1385, 50521, 663061, 5162421, 28862471, 127838711, 475638163, 1544454483, 4494470838, 11949575638, 29449955678, 68035028126, 148639284066, 309297261826, 616573557226, 1183184394986, 2194487337735, 3947417571735, 6906579371835, 11783600663835, 19647572529585

Offset: 0

Wolfdieter Lang, Mar 16 2001

Fourfold iterated sums of squares; see A060061 for threefold case.

T. D. Noe, Table of n, a(n) for n = 0..1000

CoefficientList[Series[(1385+32516*x+114318*x^2+87156*x^3+11025*x^4)/(1-x)^13 ,{x,0,20}],x] (* Indranil Ghosh, Feb 21 2017 *)

G.f. (1385+32516*x+114318*x^2+87156*x^3+11025*x^4)/(1-x)^13 = p(4, x)/(1-x)^(4*3+1) with p(2, x)=sum(A060063(4, m)*x^m, m=0..4).

cp's OEIS Frontend

A091742 Second column (m=1) of triangle A060063.

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A091743 Third column (m=2) of triangle A060063 divided by 9.

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A091744 Fourth column (m=3) of triangle A060063 divided by 9.

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A091745 Alternating row sums of triangle A060063.

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A052502 Number of permutations sigma of [3n] without fixed points such that sigma^3 = Id.

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A060058 Triangle of numbers related to A000330 (sum of squares) and A000364 (Euler numbers).

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A060061 Fourth column of triangle A060058.

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A060060 Third column of triangle A060058.

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A060062 Fifth column of triangle A060058.

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