A177747 -id:A177747 - cp's OEIS frontend

A210569 a(n) = (n-3)(n-2)(n-1)n(n+1)/30.

0, 0, 0, 0, 4, 24, 84, 224, 504, 1008, 1848, 3168, 5148, 8008, 12012, 17472, 24752, 34272, 46512, 62016, 81396, 105336, 134596, 170016, 212520, 263120, 322920, 393120, 475020, 570024, 679644, 805504, 949344, 1113024, 1298528, 1507968, 1743588, 2007768, 2303028

Offset: 0

Bruno Berselli, Mar 23 2012

The following sequences are provided by the formula n*binomial(n,k) - binomial(n,k+1) = k*binomial(n+1,k+1):
. A000217(n)   for k=1,
. A007290(n+1) for k=2,
. A050534(n)   for k=3,
. a(n)         for k=4,
. A000910(n+1) for k=5.
Sum of reciprocals of a(n), for n>3: 5/16.
From a(2), convolution of oblong numbers (A002378) with themselves. - Bruno Berselli, Oct 24 2016

Bruno Berselli, Table of n, a(n) for n = 0..1000
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 3. - _N. J. A. Sloane_, Mar 23 2014
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000217, A000389, A000910, A002378, A007290, A050534, A177747.

First differences are in A033488.

Magma
```
[4*Binomial(n+1,5): n in [0..38]];
```

f:=n->(n^5-5*n^3+4*n)/30;
[seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 23 2014

LinearRecurrence[{6,-15,20,-15,6,-1}, {0,0,0,0,4,24}, 39]
CoefficientList[Series[4x^4/(1-x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
Times@@@Partition[Range[-3,40],5,1]/30 (* Harvey P. Dale, Sep 19 2020 *)

makelist(coeff(taylor(4*x^4/(1-x)^6, x, 0, n), x, n), n, 0, 38);

a(n)=(n-3)*(n-2)*(n-1)*n*(n+1)/30 \\ Charles R Greathouse IV, Oct 07 2015

[4*binomial(n+1,5) for n in (0..40)] # G. C. Greubel, May 23 2022

G.f.: 4*x^4/(1-x)^6.
a(n) = n*binomial(n,4)-binomial(n,5) = 4*binomial(n+1,5) = 4*A000389(n+1).
a(n) = 2*A177747(2*n-7), with A177747(-7) = A177747(-5) = A177747(-3) = A177747(-1) = 0.
(n-4)*a(n) = (n+1)*a(n-1).
E.g.f.: (1/30)*x^4*(5+x)*exp(x). - G. C. Greubel, May 23 2022
Sum_{n>=4} (-1)^n/a(n) = 20*log(2) - 655/48. - Amiram Eldar, Jun 02 2022

A157901 Triangle read by rows: A000012 * A157898.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 3, 6, 10, 8, 8, 3, 9, 16, 24, 16, 16, 4, 12, 28, 40, 56, 32, 32, 4, 16, 40, 80, 96, 128, 64, 64, 5, 20, 60, 120, 216, 224, 288, 128, 128, 5, 25, 80, 200, 336, 560, 512, 640, 256, 256, 6, 30, 110, 280, 616, 896, 1408, 1152, 1408, 512, 512

Offset: 0

Gary W. Adamson & Roger L. Bagula, Mar 08 2009

Multiplication of the lower triangular matrix A157898 from the left by A000012 means: these are partial column sums of A157898.

			First few rows of the triangle, n>=0:
  1;
  1,  1;
  2,  2,  2;
  2,  4,  4,   4;
  3,  6, 10,   8,   8;
  3,  9, 16,  24,  16,  16;
  4, 12, 28,  40,  56,  32,  32;
  4, 16, 40,  80,  96, 128,  64,  64;
  5, 20, 60, 120, 216, 224, 288, 128, 128;
  5, 25, 80, 200, 336, 560, 512, 640, 256, 256;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000012, A105635, A157898.
Columns: A004526 (k=0), A002620 (k=1), A006584 (k=2), 4*A096338 (k=3), 8*A177747 (k=4), 16*A299337 (k=5), 32*A178440 (k=6).
Sums include: A105635(n+1) (row), A166486(n+1) (alternating sign diagonal), A232801(n+1) (diagonal).

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function t(n, k) // t = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
  end if; return t;
end function;
A157898:= func< n, k | (&+[(-1)^(n-j)*Binomial(n, j)*t(j, k): j in [k..n]]) >;
A157071:= func< n,k | (&+[A157898(j+k,k): j in [0..n-k]]) >;
[A157071(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 27 2025

t[n_, k_]:= t[n,k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1,k-1] +2*t[n-1,k] - (2 -Boole[n==2])*t[n-2,k-1]]; (* t = A059576 *)
A157898[n_, k_]:= A157898[n,k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
A157901[n_, k_]:= A157901[n,k]= Sum[A157898[j+k,k], {j,0,n-k}];
Table[A157901[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 27 2025 *)

@CachedFunction
def t(n, k): # t = A059576
    if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
    else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n, k): return sum((-1)^(n+k-j)*binomial(n, j+k)*t(j+k, k) for j in range(n-k+1))
def A157071(n,k): return sum(A157898(j+k,k) for j in range(n-k+1))
print(flatten([[A157071(n,k) for k in range(n+1)] for n in range(10)])) # G. C. Greubel, Aug 27 2025

T(n,k) = Sum_{j=0..n} A157898(j,k).

Edited by the Associate Editors of the OEIS, Apr 10 2009

More terms from G. C. Greubel, Aug 27 2025

cp's OEIS Frontend

A210569 a(n) = (n-3)(n-2)(n-1)n(n+1)/30.

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A157901 Triangle read by rows: A000012 * A157898.

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cp's OEIS Frontend

A210569 a(n) = (n-3)*(n-2)*(n-1)*n*(n+1)/30.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

SageMath

Formula

A157901 Triangle read by rows: A000012 * A157898.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A210569 a(n) = (n-3)(n-2)(n-1)n(n+1)/30.