cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A168598 G.f.: exp( Sum_{n>=1} A002426(n)^2*x^n/n ), where A002426(n) is the central trinomial coefficients.

Original entry on oeis.org

1, 1, 5, 21, 119, 703, 4515, 30227, 210274, 1503930, 11008198, 82099262, 622013122, 4775754930, 37089503826, 290914775618, 2301706690657, 18351027768401, 147308337621061, 1189704370416949, 9661185599013209, 78844977025403657
Offset: 0

Views

Author

Paul D. Hanna, Dec 01 2009

Keywords

Comments

Compare to: exp( Sum_{n>=1} A002426(n)*x^n/n ) = g.f. of the Motzkin numbers (A001006).

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 119*x^4 + 703*x^5 +...
log(A(x)) = x + 9*x^2/2 + 49*x^3/3 + 361*x^4/4 + 2601*x^5/5 + 19881*x^6/6 +...+ A002426(n)^2*x^n/n +...

Links

Crossrefs

Cf. A001006, A002426, A168597, A168599.

Programs

  • Magma
    m:=30;
A002426:= func< n | (&+[ Binomial(n, k)*Binomial(k, n-k): k in [0..n]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( Exp( (&+[A002426(j)^2*x^j/j: j in [1..m+2]]) ) )); // G. C. Greubel, Mar 16 2021
  • Mathematica
    A002426[n_]:= GegenbauerC[n, -n, -1/2];
With[{m=30}, CoefficientList[Series[Exp[Sum[A002426[j]^2*x^j/j, {j, m+2}]], {x, 0, m}], x]] (* G. C. Greubel, Mar 16 2021 *)
  • PARI
    {a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,polcoeff((1+x+x^2)^m,m)^2*x^m/m)+x*O(x^n)),n))}
  • Sage
    m=30
def A002426(n): return sum( binomial(n, k)*binomial(k, n-k) for k in (0..n) )
def A168598_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp( sum( A002426(j)^2*x^j/j for j in [1..m+2])) ).list()
A168598_list(m) # G. C. Greubel, Mar 16 2021

A225328 a(n) = A002426(n)^n, where A002426 is the central trinomial coefficients.

Original entry on oeis.org

1, 1, 9, 343, 130321, 345025251, 7858047974841, 1447930954097073657, 2255178731296086753063201, 29588424532574699588724679418659, 3308916781795356089160906125431831800049, 3166064605712293355286523525163381509588445189997
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2013

Keywords

Comments

Logarithmic derivative of A168599 (upon ignoring the initial term, a(0), of this sequence).

Examples

			L.g.f.: L(x) = x + 9*x^2/2 + 343*x^3/3 + 130321*x^4/4 + 345025251*x^5/5 + ...
where exponentiation is an integer series:
exp(L(x)) = 1 + x + 5*x^2 + 119*x^3 + 32707*x^4 + 69038213*x^5 + 1309743837515*x^6 + ... + A168599(n)*x^n + ...

Links

Crossrefs

Cf. A168599, A002426.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, 3^n Hypergeometric2F1[1/2, -n, 1, 4/3]]; Table[a[n]^n, {n, 0, 50}] (* G. C. Greubel, Feb 27 2017 *)
  • PARI
    {a(n)=sum(k=0,n, binomial(n, k)*binomial(k, n-k))^n}
for(n=0,20,print1(a(n),", "))

Formula

L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=0} A168599(n)*x^n ).

A322135 Table of truncated square pyramid numbers, read by antidiagonals.

Original entry on oeis.org

1, 4, 5, 9, 13, 14, 16, 25, 29, 30, 25, 41, 50, 54, 55, 36, 61, 77, 86, 90, 91, 49, 85, 110, 126, 135, 139, 140, 64, 113, 149, 174, 190, 199, 203, 204, 81, 145, 194, 230, 255, 271, 280, 284, 285, 100, 181, 245, 294, 330, 355, 371, 380, 384, 385, 121, 221, 302
Offset: 1

Views

Author

Allan C. Wechsler, Nov 27 2018

Keywords

Comments

The n-th row contains n numbers: n^2, n^2 + (n-1)^2, ..., n^2 + (n-1)^2 + ... + 1^2.
All numbers that appear in the table are listed in ascending order at A034705.
All numbers that appear twice or more are listed at A130052.
The left column is A000290 (the squares).
The top row is A000330 (the square pyramidal numbers).
The columns are A000290, A099776 (or a tail of A001844), a tail of A005918 or A120328, a tail of A027575, a tail of A027578, a tail of A027865, ...
The first two rows are A000330 and a tail of A168599, but subsequent rows are not currently in the OEIS, and are all tails of A000330 minus various constants.
The main diagonal is A050410.

Examples

			The 17th term is entry 2 on antidiagonal 6, so we sum two terms: 6^2 + 5^2 = 61.
Table begins:
   1   5  14  30  55  91 140 204 ...
   4  13  29  54  90 139 203 ...
   9  25  50  86 135 199 ...
  16  41  77 126 190 ...
  25  61 110 174 ...
  36  85 149 ...
  49 113 ...
  64 ...
  ...

Crossrefs

See comments; also cf. A000330, A059255.

Programs

  • Mathematica
    T[n_,k_] = Sum[(n+i)^2, {i,0,k-1}]; Table[T[n-k+1, k], {n,1,10},  {k,1,n}] // Flatten (* Amiram Eldar, Nov 28 2018 *)
f[n_] := Table[SeriesCoefficient[-((y (y (1 + y) + x (1 - 2 y - 3 y^2) + x^2 (1 - 3 y + 4 y^2)))/((-1 + x)^3 (-1 + y)^4)) , {x, 0,
i + 1 - j}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f, 10]] (* Stefano Spezia, Nov 28 2018 *)

Formula

T(n,k) = n^2 + (n+1)^2 + ... + (n+k-1)^2 = A000330(n + k - 1) - A000330(n - 1) = T(n, k) = k*n^2 + (k^2 - k)*n + (1/3*k^3 - 1/2*k^2 + 1/6*k)
G.f.: -y*(y*(1 + y) + x*(1 - 2*y - 3*y^2) + x^2*(1 - 3*y + 4*y^2))/((- 1 + x)^3*(- 1 + y)^4). - Stefano Spezia, Nov 28 2018
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.