A257055 -id:A257055 - cp's OEIS frontend

A256859 a(n) = n(n + 1)(n + 2)*(n^2 - n + 4)/24.

1, 6, 25, 80, 210, 476, 966, 1800, 3135, 5170, 8151, 12376, 18200, 26040, 36380, 49776, 66861, 88350, 115045, 147840, 187726, 235796, 293250, 361400, 441675, 535626, 644931, 771400, 916980, 1083760, 1273976, 1490016, 1734425, 2009910, 2319345, 2665776, 3052426

Offset: 1

Luciano Ancora, Apr 14 2015

This is the case k = n of b(n,k) = n*(n+1)*(n+2)*(k*(n-1)+4)/24, where b(n,k) is the n-th hypersolid number in 4 dimensions generated from an arithmetical progression with the first term 1 and common difference k. Therefore, the sequence is the main diagonal of the Table 3 in Sardelis et al. paper (see Links field).

G. C. Greubel, Table of n, a(n) for n = 1..1000
D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070v1 [math.GM], 2008.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389, A261721.

Cf. similar sequences of the form binomial(n+k-2,k-1)+n*binomial(n+k-2,k): A006000 (k=2); A257055 (k=3); this sequence (k=4); A256860 (k=5); A256861 (k=6).

[n*(n + 1)*(n + 2)*(n^2 - n + 4)/24: n in [1..30]]; // G. C. Greubel, Nov 23 2017

Table[n (n + 1) (n + 2) (n^2 - n + 4)/24, {n, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,6,25,80,210,476},40] (* Harvey P. Dale, Mar 19 2022 *)

vector(40, n, n*(n+1)*(n+2)*(n^2-n+4)/24) \\ Bruno Berselli, Apr 15 2015

G.f.: x*(1 + 4*x^2)/(1 - x)^6.
a(n) = 4*A000389(n+2) + A000389(n+4). - Bruno Berselli, Apr 15 2015
E.g.f.: (24*x + 48*x^2 + 40*x^3 + 12*x^4 + x^5)*exp(x)/24. - G. C. Greubel, Nov 23 2017
a(n) = A261721(n,n-1). - Alois P. Heinz, Apr 15 2020

A208657 Triangular array read by rows: n*binomial(n,n-k+1)-binomial(n-1,n-k) with k = 1..n, n >= 1.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 3, 13, 21, 15, 4, 21, 44, 46, 24, 5, 31, 80, 110, 85, 35, 6, 43, 132, 225, 230, 141, 48, 7, 57, 203, 413, 525, 427, 217, 63, 8, 73, 296, 700, 1064, 1078, 728, 316, 80, 9, 91, 414, 1116, 1974, 2394, 2016, 1164, 441, 99, 10, 111, 560, 1695

Offset: 1

Clark Kimberling, Mar 01 2012

Mirror of A208656.

			Triangle begins:
0,
1, 3,
2, 7, 8,
3, 13, 21, 15,
4, 21, 44, 46, 24,
5, 31, 80, 110, 85, 35,
6, 43, 132, 225, 230, 141, 48,
7, 57, 203, 413, 525, 427, 217, 63,
8, 73, 296, 700, 1064, 1078, 728, 316, 80,
9, 91, 414, 1116, 1974, 2394, 2016, 1164, 441, 99;
...

Cf. A002061 (second column), A208656, A208658 (row sums), A257055.

[n*Binomial(n,n-k+1)-Binomial(n-1,n-k): k in [1..n], n in [1..11]]; // Bruno Berselli, Apr 15 2015

z = 12;
f[n_, k_] := n*Binomial[n, k] - Binomial[n - 1, k - 1]
t = Table[f[n, k], {n, 1, z}, {k, 1, n}];
TableForm[t] (* A208656 as a triangle *)
Flatten[t]   (* A208656 as a sequence *)
r = Table[f[n, k], {n, 1, z}, {k, n, 1, -1}];
TableForm[r] (* A208657 as a triangle *)
Flatten[r]   (* A208657 as a sequence *)
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 3 z}](* A208658 *)

cp's OEIS Frontend

A256859 a(n) = n(n + 1)(n + 2)*(n^2 - n + 4)/24.

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A208657 Triangular array read by rows: n*binomial(n,n-k+1)-binomial(n-1,n-k) with k = 1..n, n >= 1.

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cp's OEIS Frontend

A256859 a(n) = n*(n + 1)*(n + 2)*(n^2 - n + 4)/24.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A208657 Triangular array read by rows: n*binomial(n,n-k+1)-binomial(n-1,n-k) with k = 1..n, n >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

A256859 a(n) = n(n + 1)(n + 2)*(n^2 - n + 4)/24.