A249349 -id:A249349 - cp's OEIS frontend

A249354 a(n) = n(3n^2 + 3*n + 1).

0, 7, 38, 111, 244, 455, 762, 1183, 1736, 2439, 3310, 4367, 5628, 7111, 8834, 10815, 13072, 15623, 18486, 21679, 25220, 29127, 33418, 38111, 43224, 48775, 54782, 61263, 68236, 75719, 83730, 92287, 101408, 111111, 121414, 132335, 143892, 156103, 168986

Offset: 0

M. F. Hasler, Oct 26 2014

The series Sum a(n)/A007559(n+1)^3 has the sum 1/9 (cf. A249352), analogous to Sum_{n=1..oo} A000217(n)/A001147(n+1)^2 = 1/8 (cf. A249348 and A249349).

Also, nonnegative numbers m such that 9*m + 1 is a cube. - Bruno Berselli, May 23 2017

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A132355: numbers m such that 9*m + 1 is a square.

Table[n (3 n^2 + 3 n + 1), {n, 0, 38}] (* or *)
CoefficientList[Series[x (7 + 10 x + x^2)/(x - 1)^4, {x, 0, 38}], x] (* Michael De Vlieger, May 23 2017 *)

PARI
```
A249354(n)=3*n^3+3*n^2+n
```

G.f.: x*(7+10*x+x^2) / (x-1)^4 . - R. J. Mathar, Oct 28 2014

a(n) = n*A003215(n). - R. J. Mathar, Oct 28 2014

A350557 Triangle T(n,k) read by rows with T(n,0) = (2n)! / (2^n n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2i)!) (2n)! / (2^n n!) for 0 < k <= n.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 15, 21, 7, 1, 105, 148, 52, 10, 1, 945, 1333, 472, 96, 13, 1, 10395, 14664, 5197, 1066, 153, 16, 1, 135135, 190633, 67567, 13873, 2009, 223, 19, 1, 2027025, 2859496, 1013512, 208116, 30170, 3380, 306, 22, 1

Offset: 0

Werner Schulte, Jan 05 2022

			Triangle T(n,k) for 0 <= k <= n starts:
n\k :        0        1        2       3      4     5    6   7  8
=================================================================
  0 :        1
  1 :        1        1
  2 :        3        4        1
  3 :       15       21        7       1
  4 :      105      148       52      10      1
  5 :      945     1333      472      96     13     1
  6 :    10395    14664     5197    1066    153    16    1
  7 :   135135   190633    67567   13873   2009   223   19   1
  8 :  2027025  2859496  1013512  208116  30170  3380  306  22  1
  etc.

Cf. A001147 (column 0), A286286 (column 1), A249349 (column 2).
Cf. A000007 (alternating row sums).
Cf. A350512.

Flatten[Table[If[k==0,(2n)!/(2^n n!),Sum[Binomial[i-1,k-1]2^i i!/(2i)!,{i,k,n}](2n)!/(2^n n!)],{n,0,8},{k,0,n}]] (* Stefano Spezia, Jan 06 2022 *)

T(n,n) = 1.
T(n,k) = binomial(n-1,k-1) + (2*n - 1) * T(n-1,k) for 0 < k < n.
Conjecture: M(n,k) = (-1)^(n-k) * T(n,k) is matrix inverse of A350512.

cp's OEIS Frontend

A249354 a(n) = n(3n^2 + 3*n + 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A350557 Triangle T(n,k) read by rows with T(n,0) = (2n)! / (2^n n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2i)!) (2n)! / (2^n n!) for 0 < k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A249354 a(n) = n*(3*n^2 + 3*n + 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A350557 Triangle T(n,k) read by rows with T(n,0) = (2*n)! / (2^n * n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2*i)!) * (2*n)! / (2^n * n!) for 0 < k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A249354 a(n) = n(3n^2 + 3*n + 1).

A350557 Triangle T(n,k) read by rows with T(n,0) = (2n)! / (2^n n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2i)!) (2n)! / (2^n n!) for 0 < k <= n.