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.

A138247 E.g.f.: Sum_{n>=0} exp( (2^n+3^n)*x ) * (2^n+3^n)^n * x^n/n!.

Original entry on oeis.org

1, 7, 223, 49849, 94705663, 1616229320497, 251286598125520183, 357716675257916544062689, 4670472774542449929397808845183, 559006854195449142958954163012808059617, 612171730457531439763516750114999086563829844663, 6118056385739077528636842573416061383741677666682643900049
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2008, revised Mar 11 2008

Keywords

Comments

GENERAL BINOMIAL IDENTITY.
When p=2, q=3, this sequence illustrates the following identity:
Sum_{k=0..n} C(n,k)*(p^k + q^k)^n =
Sum_{k=0..n} C(n,k)*(1 + p^(n-k)*q^k)^n
which is a special case of the more general binomial identity:
Sum_{k=0..n} C(n,k)*(s*p^k + t*q^k)^(n-k) * (u*p^k + v*q^k)^k =
Sum_{k=0..n} C(n,k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k.

Examples

			E.g.f.: A(x) = 1 + 7*x + 223*x^2/2! + 49849*x^3/3! + 94705663*x^4/4! + 1616229320497*x^5/5! + 251286598125520183*x^6/6! + 357716675257916544062689*x^7/7! + 4670472774542449929397808845183*x^8/8! + ...
such that
A(x) = exp(2*x) + (2+3)*exp((2+3)*x)*x + (2^2+3^2)^2*exp((2^2+3^2)*x)*x^2/2! + (2^3+3^3)^3*exp((2^3+3^3)*x)*x^3/3! + (2^4+3^4)^4*exp((2^4+3^4)*x)*x^4/4! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 7*x + 223*x^2 + 49849*x^3 + 94705663*x^4 + 1616229320497*x^5 + 251286598125520183*x^6 + 357716675257916544062689*x^7 + ...
such that
B(x) = 1/(1-2*x) + (2+3)*x/(1 - (2+3)*x)^2 + (2^2+3^2)^2*x^2/(1 - (2^2+3^2)*x)^3 + (2^3+3^3)^3*x^3/(1 - (2^3+3^3)*x)^4 + (2^4+3^4)^4*x^4/(1 - (2^4+3^4)*x)^5 + ...
ILLUSTRATION OF TERMS.
a(1) = 2 + 5 = 3 + 4 = 7 ;
a(2) = 2^2 + 2*5^2 + 13^2 = 5^2 + 2*7^2 + 10^2 = 223 ;
a(3) = 2^3 + 3*5^3 + 3*13^3 + 35^3 = 9^3 + 3*13^3 + 3*19^3 + 28^3 = 49849.

Crossrefs

Cf. A007689, A094617, A196457, A196458, A196459, A196460.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*(2^k + 3^k)^n, {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Jul 14 2019 *)
  • PARI
    {a(n)=local(p=2,q=3,s=1,t=1,u=1,v=1);
sum(k=0,n,binomial(n,k)*(s*p^k + t*q^k)^(n-k)*(u*p^k + v*q^k)^k)}
/* right side of the general binomial identity: */
{a(n)=local(p=2,q=3,s=1,t=1,u=1,v=1);
sum(k=0,n,binomial(n,k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k)}

Formula

E.g.f.: Sum_{n>=0} (2^n + 3^n)^n * exp( (2^n + 3^n)*x ) * x^n / n!.
O.g.f.: Sum_{n>=0} (2^n + 3^n)^n * x^n / (1 - (2^n + 3^n)*x)^(n+1). - Paul D. Hanna, Jul 13 2019
FORMULAS FOR TERMS.
a(n) = Sum_{k=0..n} C(n,k)*(2^k + 3^k)^n.
a(n) = Sum_{k=0..n} C(n,k)*(1 + 2^(n-k)*3^k)^n.
a(n) = Sum_{k=0..n} C(n,k)*A007689(k)^n.
a(n) = Sum_{k=0..n} C(n,k)*A094617(n,k)^n.
a(n) ~ 3^(n^2). - Vaclav Kotesovec, Jul 14 2019

A094615 Triangular array T of numbers generated by these rules: 1 is in T; and if x is in T, then 2x+1 and 3x+2 are in T.

Original entry on oeis.org

1, 3, 5, 7, 11, 17, 15, 23, 35, 53, 31, 47, 71, 107, 161, 63, 95, 143, 215, 323, 485, 127, 191, 287, 431, 647, 971, 1457, 255, 383, 575, 863, 1295, 1943, 2915, 4373, 511, 767, 1151, 1727, 2591, 3887, 5831, 8747, 13121, 1023, 1535, 2303, 3455, 5183, 7775, 11663, 17495, 26243, 39365
Offset: 0

Views

Author

Clark Kimberling, May 14 2004

Keywords

Comments

To obtain row n from row n-1, apply 2x+1 to each x in row n-1 and then put -1+2*3^n at the end. Or, instead, apply 3x+2 to each x in row n-1 and then put -1+2^(n+1) at the beginning.
Subtriangle of the triangle in A230445. - Philippe Deléham, Oct 31 2013

Examples

			Triangle begins:
  n\k|   1    2    3    4    5    6     7
  ---+-----------------------------------
  0  |   1;
  1  |   3,   5;
  2  |   7,  11,  17;
  3  |  15,  23,  35,  53;
  4  |  31,  47,  71, 107, 161;
  5  |  63,  95, 143, 215, 323, 485;
  6  | 127, 191, 287, 431, 647, 971, 1457;

Links

Crossrefs

Cf. A094616 (row sums), A094617, A230445.
Cf. A048473, A171498, A198644

Programs

  • PARI
    tabl(nn) = {my(row = [1], nrow); for (n=1, nn, print (row); nrow = vector(n+1, k, if (k<=n, (2*row[k]+1), -1+2*3^n)); row = nrow;);} \\ Michel Marcus, Nov 14 2020

Formula

T(n,0) = -1+2^(n+1) = A000225(n+1).
T(n,n) = -1+2*3^n = A048473(n).
T(2n,n) = -1+2*6^n.
T(n,k) = -1 + 2^(n+1-k)*3^k. - Lamine Ngom, Feb 10 2021

Extensions

Offset 0 and more terms from Michel Marcus, Nov 14 2020

A094618 a(n) = 3^(n+1) - 2^(n+1) + n + 1.

Original entry on oeis.org

2, 7, 22, 69, 216, 671, 2066, 6313, 19180, 58035, 175110, 527357, 1586144, 4766599, 14316154, 42981201, 129009108, 387158363, 1161737198, 3485735845, 10458256072, 31376865327, 94134790242, 282412759289, 847255055036, 2541798719491, 7625463267286, 22876524019533
Offset: 0

Views

Author

Clark Kimberling, May 14 2004

Keywords

Comments

Row sums of A094617.

Links

Crossrefs

Cf. A094617.

Programs

  • Maple
    seq(3^(n+1) - 2^(n+1) + n + 1, n=0..100); # Robert Israel, Jul 22 2018
  • Mathematica
    Table[3^(n+1)-2^(n+1)+n+1,{n,0,30}] (* or *) LinearRecurrence[{7,-17,17,-6},{2,7,22,69},30] (* Harvey P. Dale, Oct 11 2022 *)
  • PARI
    a(n) = 3^(n+1) - 2^(n+1) + n + 1; \\ Michel Marcus, Jun 05 2016

Formula

a(n) = 2*a(n-1) + 1 - n + 3^n, a(0) = 2.
G.f.: (2-7*x+7*x^2)/(1-7*x+17*x^2-17*x^3+6*x^4). - Robert Israel, Jul 22 2018
From Elmo R. Oliveira, Mar 06 2025: (Start)
E.g.f.: exp(x)*(1 + x + 3*exp(2*x) - 2*exp(x)).
a(n) = 7*a(n-1) - 17*a(n-2) + 17*a(n-3) - 6*a(n-4). (End)

Extensions

New definition from Ralf Stephan, Dec 01 2004
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