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.

A303921 Main diagonal of triangle A303920: a(n) = A303920(n,n) for n>=0.

Original entry on oeis.org

1, 1, 6, 56, 722, 12012, 246092, 6002824, 170048394, 5489377628, 198966923232, 8002061191632, 353657146741108, 17038311744899928, 888756685396257456, 49903123853737160256, 3001090647251938886634, 192456294604677056842812, 13110208254597852188752232, 945417747582856587884200944, 71952514694665595216762956518, 5763451519600988678663191769380
Offset: 0

Views

Author

Paul D. Hanna, May 02 2018

Keywords

Comments

G.f. of A303920: (1-y) * Sum_{n>=0} y^n * (1 + x*(1-y)^2)^(n^2) = Sum_{n>=0} Sum_{k=0..2*n} A303920(n,k)*x^n*y^k; this sequence is A303920(n,n) for n>=0.

Examples

			Triangle A303920 begins:
[1];
[0, 1, 1];
[0, 0, 6, 6, 0];
[0, 0, 4, 56, 56, 4, 0];
[0, 0, 1, 117, 722, 722, 117, 1, 0];
[0, 0, 0, 126, 2982, 12012, 12012, 2982, 126, 0, 0];
[0, 0, 0, 84, 6916, 79548, 246092, 246092, 79548, 6916, 84, 0, 0]; ...
the main diagonal of which forms this sequence; note that the row sums of A303920 equals A001813(n) = (2*n)!/n!.

Links

Crossrefs

Cf. A303920, A303922.

Formula

a(n) ~ sqrt(3) * 2^(2*n + 1/2) * n^(n - 1/2) / (sqrt(Pi) * exp(n)). - Vaclav Kotesovec, Oct 08 2019

A303922 Column sums of triangle A303920.

Original entry on oeis.org

1, 1, 12, 435, 60607, 32465376, 67856416808, 560418604644648, 18418643482653787248, 2416653303692582729686744, 1267452375341631770930186428169, 2658327966985973593187656395635032767, 22300420873364447640210289607043443823426176, 748285604725151189853520504436684719836490370604576
Offset: 0

Views

Author

Paul D. Hanna, May 02 2018

Keywords

Comments

G.f. of A303920: (1-y) * Sum_{n>=0} y^n * (1 + x*(1-y)^2)^(n^2) = Sum_{n>=0} Sum_{k=0..2*n} A303920(n,k)*x^n*y^k; the g.f. of this sequence is at y=x, x=1.

Examples

			G.f.: A(x) = 1 + x + 12*x^2 + 435*x^3 + 60607*x^4 + 32465376*x^5 + 67856416808*x^6 + 560418604644648*x^7 + 18418643482653787248*x^8 + ...
such that
A(x)/(1-x) = 1 + x*(2 - 2*x + x^2) + x^2*(2 - 2*x + x^2)^4 + x^3*(2 - 2*x + x^2)^9 + x^4*(2 - 2*x + x^2)^16 + x^5*(2 - 2*x + x^2)^25 + ...

Links

Crossrefs

Cf. A303920, A303921.

Programs

  • PARI
    /* G.f. by Definition: */
{a(n) = my(A = (1-x) * sum(m=0,2*n, x^m * (1 + (1-x)^2  +x*O(x^n) )^(m^2))); polcoeff(A, n,x)}
for(n=0, 20, print1(a(n),", "))
  • PARI
    /* Continued fraction expression: */
{a(n) = my(CF=1, q = 1 + (1-x)^2 +x*O(x^n)); for(k=0, n, CF = 1/(1 - q^(4*n-4*k+1)*x/(1 - q^(2*n-2*k+1)*(q^(2*n-2*k+2) - 1)*x*CF)) ); polcoeff((1-x)*CF, n,x)}
for(n=0, 20, print1(a(n),", "))
  • PARI
    /* G.f. by q-series identity: */
{a(n) = my(A =1, q = 1 + (1-x)^2 +x*O(x^n)); A = (1-x) * sum(m=0,2*n, x^m*q^m * prod(k=1,m, (1 - x*q^(4*k-3)) / (1 - x*q^(4*k-1) +x*O(x^n)) )); polcoeff(A, n,x)}
for(n=0, 20, print1(a(n),", "))

Formula

GENERATING FUNCTIONS.
(1) A(x) = (1-x) * Sum_{n>=0} x^n * (1 + (1-x)^2)^(n^2).
(2) A(x) = (1-x) * Sum_{n>=0} x^n*q^n * Product_{k=1..n} (1 - q^(4*k-3)*x) / (1 - q^(4*k-1)*x) where q = 1 + (1-x)^2, due to a q-series identity.
(3) A(x) = (1-x)/(1 - q*x/(1 - q*(q^2-1)*x/(1 - q^5*x/(1 - q^3*(q^4-1)*x/(1 - q^9*x/(1- q^5*(q^6-1)*x/(1 - q^13*x/(1 - q^7*(q^8-1)*x/(1 - ...))))))))) where q = 1 + (1-x)^2, a continued fraction due to an identity of a partial elliptic theta function.
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Oct 08 2019

A265936 G.f.: Sum_{n>=0} (1 + x)^(n^2) / 2^n.

Original entry on oeis.org

2, 6, 72, 1488, 43212, 1615824, 73897824, 3995603040, 249332628600, 17635891224600, 1394325697514112, 121850733102557184, 11663364820483368384, 1213527023075625127296, 136368036713802512640384, 16459661773011642351224832, 2123742016843422531580031760, 291703805646180152870305600416, 42495460119330209128505618419584, 6544578588779477399509681497008256, 1062399800520315889891506552001161024, 181308080907736435566683700136306288320
Offset: 0

Views

Author

Paul D. Hanna, Dec 23 2015

Keywords

Examples

			G.f.: A(x) = 2 + 6*x + 72*x^2 + 1488*x^3 + 43212*x^4 + 1615824*x^5 + 73897824*x^6 + 3995603040*x^7 + 249332628600*x^8 + 17635891224600*x^9 +...
where
A(x) = 1 + (1+x)/2 + (1+x)^4/2^2 + (1+x)^9/2^3 + (1+x)^16/2^4 + (1+x)^25/2^5 + (1+x)^36/2^6 + (1+x)^49/2^7 + (1+x)^64/2^8 +...+ (1+x)^(n^2)/2^n +...

Links

Crossrefs

Cf. A265937, A303920, A173217.

Programs

  • Mathematica
    Table[Round[Sum[Binomial[k^2, n]/2^k, {k, Sqrt[n], Infinity}]] , {n, 0, 20}] (* G. C. Greubel, May 23 2017 *)
Table[2*Sum[StirlingS1[n, j] * HurwitzLerchPhi[1/2, -2*j, 0]/2, {j, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2019 *)
  • PARI
    /* Informal listing of terms: */
{Vec( round( sum(n=0,600,(1+x +O(x^31))^(n^2)/2^n * 1.) ) )}
{Vec( round( sum(n=0,200, (1.+x)^n/2^n * prod(k=1,n, (2 - (1+x)^(4*k-3)) / (2 - (1+x)^(4*k-1)) +O(x^21) ) ) ) )}

Formula

G.f.: Sum_{n>=0} (1+x)^n/2^n * Product_{k=1..n} (2 - (1+x)^(4*k-3)) / (2 - (1+x)^(4*k-1)) due to a q-series identity.
G.f.: 1/(1 - (1+x)/2 /(1 - (1+x)*((1+x)^2-1)/2 /(1 - (1+x)^5/2 /(1 - (1+x)^3*((1+x)^4-1)/2 /(1 - (1+x)^9/2 /(1 - (1+x)^5*((1+x)^6-1)/2 /(1 - (1+x)^13/2 /(1 - (1+x)^7*((1+x)^8-1)/2 /(1 - ...))))))))), a continued fraction due to a partial elliptic theta function identity.
a(n) = Sum_{k>=sqrt(n)} binomial(k^2,n) / 2^k.
a(n) = Sum_{k=0..2*n} A303920(n,k) * 2^k, for n>0.
a(n) = 2 * A173217(n) for n>=0.
a(n) ~ 2^(2*n + 1/2 - log(2)/8) * n^n / (exp(n) * log(2)^(2*n + 1)). - Vaclav Kotesovec, Oct 08 2019
Showing 1-3 of 3 results.

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