A208657 -id:A208657 - cp's OEIS frontend

A208658 Row sums of A208657.

0, 4, 17, 52, 139, 346, 825, 1912, 4343, 9718, 21493, 47092, 102387, 221170, 475121, 1015792, 2162671, 4587502, 9699309, 20447212, 42991595, 90177514, 188743657, 394264552, 822083559, 1711276006, 3556769765, 7381975012

Offset: 1

Clark Kimberling, Mar 01 2012

Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A208657.

f[n_, k_] := n*Binomial[n, k] - Binomial[n - 1, k - 1]
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 3 z}](* A208658 *)

Conjectures from Colin Barker, Mar 20 2012: (Start)
a(n) = -2^(-1+n)+(-1+2^n)*n.
G.f.: x^2*(4-7*x+2*x^2)/((1-x)^2*(1-2*x)^2). (End)
From Stefano Spezia, Nov 19 2023: (Start)
The above conjectures are true.
E.g.f.: exp(x)*(exp(x)*(4*x - 1) - 2*x)/2. (End)

A257055 a(n) = n(n + 1)(n^2 - n + 3)/6.

Original entry on oeis.org

0, 1, 5, 18, 50, 115, 231, 420, 708, 1125, 1705, 2486, 3510, 4823, 6475, 8520, 11016, 14025, 17613, 21850, 26810, 32571, 39215, 46828, 55500, 65325, 76401, 88830, 102718, 118175, 135315, 154256, 175120, 198033, 223125, 250530, 280386, 312835, 348023, 386100

Offset: 0

Bruno Berselli, Apr 15 2015

Partial sums of A037235.
After 0, this sequence is the 2nd diagonal of the square array in A080851.
For n > 2, a(n)-n is the 4th column of the triangular array in A208657.

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

Cf. A000332, A037235, A080851, A208657.

Cf. similar sequences listed in A256859.

Magma
```
[n*(n+1)*(n^2-n+3)/6: n in [0..40]];
```

Table[n (n + 1) (n^2 - n + 3)/6, {n, 40}]

vector(40, n, n--; n*(n+1)*(n^2-n+3)/6)

Sage
```
[n*(n+1)*(n^2-n+3)/6 for n in (0..40)]
```

G.f.: x*(1 + 3*x^2)/(1 - x)^5.
a(n) = 3*A000332(n+1) + A000332(n+3).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, May 27 2021

A208656 Triangle T(n, k) = n*C(n,k) - C(n-1,k-1), 1 <= k <= n, read by rows.

Original entry on oeis.org

0, 3, 1, 8, 7, 2, 15, 21, 13, 3, 24, 46, 44, 21, 4, 35, 85, 110, 80, 31, 5, 48, 141, 230, 225, 132, 43, 6, 63, 217, 427, 525, 413, 203, 57, 7, 80, 316, 728, 1078, 1064, 700, 296, 73, 8, 99, 441, 1164, 2016, 2394, 1974, 1116, 414, 91, 9, 120, 595, 1770

Offset: 1

Clark Kimberling, Mar 01 2012

Mirror of A208657.
col 1:  A005563
col 2:  A127736
top edge:  A000027

			First five rows:
0
3....1
8....7....2
15...21...13...3
24...46...44...21...4

Cf. A208657, A208658.

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 *)

Definition amended by Georg Fischer, Feb 01 2022

cp's OEIS Frontend

A208658 Row sums of A208657.

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A257055 a(n) = n(n + 1)(n^2 - n + 3)/6.

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Magma

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Sage

Formula

A208656 Triangle T(n, k) = n*C(n,k) - C(n-1,k-1), 1 <= k <= n, read by rows.

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cp's OEIS Frontend

A208658 Row sums of A208657.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A257055 a(n) = n*(n + 1)*(n^2 - n + 3)/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A208656 Triangle T(n, k) = n*C(n,k) - C(n-1,k-1), 1 <= k <= n, read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A257055 a(n) = n(n + 1)(n^2 - n + 3)/6.