A060060 -id:A060060 - cp's OEIS frontend

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

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).

A259181 a(n) = n(n+1)(n+2)(n+3)(2n^2+6n+7)/360.

Original entry on oeis.org

0, 1, 9, 43, 147, 406, 966, 2058, 4026, 7359, 12727, 21021, 33397, 51324, 76636, 111588, 158916, 221901, 304437, 411103, 547239, 719026, 933570, 1198990, 1524510, 1920555, 2398851, 2972529, 3656233, 4466232, 5420536, 6539016, 7843528, 9358041, 11108769

Offset: 0

Luce ETIENNE, Nov 08 2015

After 0, second bisection of A129548.

This sequence is also the total number of squares of all sizes in i X i subsquares in an n X n grid, whereas A000330 simply gives the number of all sizes of squares in an n X n grid. See illustrations.

			a(0) = 0; a(1) = 1*1; a(2) = 4*1+1*5 = 9; a(3) = 9*1+4*5+1*14 = 43.

Colin Barker, Table of n, a(n) for n = 0..1000
Luce ETIENNE, illustration of initial terms
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 14.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000217, A000290, A000330, A000539, A129548.

Cf. A060060: (1/6)*Sum_{i=0..n} (i+1)*(i+2)*(2*i+3)*i^2.

vector(100, n, n--; n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360) \\ Altug Alkan, Nov 08 2015

concat(0, Vec(-x*(x+1)^2 / (x-1)^7 + O(x^100))) \\ Colin Barker, Nov 08 2015

a(n) = (1/6)*Sum_{i=0..n} (i+1)*(i+2)*(2*i+3)*(n-i)^2.
a(n) = Sum_{i=0..n} A000290(n-i)*A000330(i+1).
G.f.: x*(1 + x)^2 / (1 - x)^7. - Colin Barker, Nov 08 2015
a(n) = (A000539(n+1) - A000217(n+1))/30. - Yasser Arath Chavez Reyes, Feb 24 2024

cp's OEIS Frontend

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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A259181 a(n) = n(n+1)(n+2)(n+3)(2n^2+6n+7)/360.

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cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A060061 Fourth column of triangle A060058.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A259181 a(n) = n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A259181 a(n) = n(n+1)(n+2)(n+3)(2n^2+6n+7)/360.