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 1-5 of 5 results.

A181143 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*y^k] * x^n/n ) = Sum_{n>=0,k=0..n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 30, 85, 30, 1, 1, 55, 337, 337, 55, 1, 1, 91, 1029, 2230, 1029, 91, 1, 1, 140, 2632, 10549, 10549, 2632, 140, 1, 1, 204, 5922, 39533, 73157, 39533, 5922, 204, 1, 1, 285, 12090, 124805, 384948, 384948, 124805, 12090, 285, 1, 1
Offset: 0

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Author

Paul D. Hanna, Oct 13 2010

Keywords

Comments

Compare g.f. to that of the following triangle variants:
* Pascal's: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)*y^k] * x^n/n );
* Narayana: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*y^k] * x^n/n );
* A181144: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*y^k] * x^n/n );
* A218115: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5*y^k] * x^n/n );
* A218116: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^6*y^k] * x^n/n ).

Examples

			G.f.: A(x,y) = 1 + (1+y)*x + (1+5*y+y^2)*x^2 + (1+14*y+14*y^2+y^3)*x^3 + (1+30*y+85*y^2+30*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^3*y + y^2)*x^2/2
+ (1 + 3^3*y + 3^3*y^2 + y^3)*x^3/3
+ (1 + 4^3*y + 6^3*y^2 + 4^3*y^3 + y^4)*x^4/4
+ (1 + 5^3*y + 10^3*y^2 + 10^3*y^3 + 5^3*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 5, 1;
1, 14, 14, 1;
1, 30, 85, 30, 1;
1, 55, 337, 337, 55, 1;
1, 91, 1029, 2230, 1029, 91, 1;
1, 140, 2632, 10549, 10549, 2632, 140, 1;
1, 204, 5922, 39533, 73157, 39533, 5922, 204, 1;
1, 285, 12090, 124805, 384948, 384948, 124805, 12090, 285, 1;
1, 385, 22869, 345389, 1648478, 2748240, 1648478, 345389, 22869, 385, 1;
1, 506, 40678, 861080, 6016297, 15525056, 15525056, 6016297, 861080, 40678, 506, 1; ...
Note that column 1 forms the sum of squares (A000330).
Inverse binomial transform of columns begins:
[1];
[1, 4, 5, 2];
[1, 13, 58, 123, 136, 76, 17];
[1, 29, 278, 1308, 3532, 5867, 6118, 3914, 1407, 218];
[1, 54, 920, 7626, 36916, 114637, 240271, 348354, 350881, 241531, 108551, 28742, 3404]; ...
the g.f. of the rightmost coefficients of which form the g.f. exp( Sum_{n>=1} (3*n)!/(3*n!^3) * x^n/n ), and yield the self-convolution of A229452.

Crossrefs

Cf. A000330 (column 1), A166990 (row sums), A166896 (antidiagonal sums), A218139.
Cf. variants: A001263 (Narayana), A181144, A218115, A218116.

Programs

  • PARI
    {T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,sum(j=0,m,binomial(m,j)^3*y^j)*x^m/m)+O(x^(n+1))),n,x),k,y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A166897 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^3*n/(n-k), n>=1.

Original entry on oeis.org

1, 3, 13, 39, 126, 477, 1765, 6495, 24709, 95128, 367368, 1431453, 5620343, 22170543, 87858813, 349708431, 1397003136, 5598513261, 22502171771, 90681323364, 366299212873, 1482827487650, 6014529069540, 24439715146941
Offset: 1

Views

Author

Paul D. Hanna, Nov 23 2009

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 39*x^4/4 + 126*x^5/5 + 477*x^6/6 +...
exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...+ A166896(n)*x^n/n +...

Crossrefs

Cf. A166897, variants: A167539, A166895, A166899.

Programs

  • Mathematica
    Table[Sum[Binomial[n-k,k]^3 n/(n-k),{k,0,Floor[n/2]}],{n,30}] (* Harvey P. Dale, Mar 05 2013 *)
  • PARI
    a(n)=sum(k=0,n\2,binomial(n-k,k)^3*n/(n-k))

Formula

Logarithmic derivative of A166896.
a(n) ~ sqrt(15) * phi^(3*n + 2) / (6*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 27 2017

A196559 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k]^2 * x^n/n ).

Original entry on oeis.org

1, 1, 3, 12, 65, 384, 2197, 14078, 94739, 670612, 4899280, 36645899, 281037158, 2197679518, 17489660228, 141241307806, 1155345218645, 9559672712389, 79905432682918, 674005489358155, 5731854529045978, 49105864505432392, 423531623342726441
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 65*x^4 + 384*x^5 + 2197*x^6 +...
where
log(A(x)) = (1 + x)^2*x + (1+2^3*x+x^2)^2*x^2/2 + (1+3^3*x+3^3*x^2+x^3)^2*x^3/3 + (1+4^3*x+6^3*x^2+4^3*x^3+x^4)^2*x^4/4 +...
More explicitly,
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 205*x^4/4 + 1506*x^5/5 + 10016*x^6/6 +...

Crossrefs

Cf. A166896, A180718, A196560.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^3*x^k)^2*x^m/m)+x*O(x^n)), n)}

A200212 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k*A(x)^(n-k)] * x^n/n ).

Original entry on oeis.org

1, 1, 3, 11, 42, 174, 763, 3457, 16075, 76351, 368767, 1805682, 8943948, 44736096, 225646033, 1146461185, 5862224756, 30144922281, 155791900727, 808773877919, 4215675455503, 22054576750972, 115765182718467, 609508331610920, 3218059655553030, 17034314889643633
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 42*x^4 + 174*x^5 + 763*x^6 +...
where the logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (A + x)*x + (A^2 + 2^3*x*A + x^2)*x^2/2 +
(A^3 + 3^3*x*A^2 + 3^3*x^2*A + x^3)*x^3/3 +
(A^4 + 4^3*x*A^3 + 6^3*x^2*A^2 + 4^3*x^3*A + x^4)*x^4/4 +
(A^5 + 5^3*x*A^4 + 10^3*x^2*A^3 + 10^3*x^3*A^2 + 5^3*x^4*A + x^5)*x^5/5 +
(A^6 + 6^3*x*A^5 + 15^3*x^2*A^4 + 20^3*x^3*A^3 + 15^3*x^4*A^2 + 6^3*x^5*A + x^6)*x^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 117*x^4/4 + 581*x^5/5 + 2987*x^6/6 + 15499*x^7/7 + 81213*x^8/8 +...

Crossrefs

Cf. A192131, A166896, A198944, A199875, A166990, A198950, A181143, A181543, A200074.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*x^j/A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}

A200215 G.f. satisfies: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^3 * x^k*A(x)^k) * x^n*A(x)^n/n ).

Original entry on oeis.org

1, 1, 3, 13, 61, 306, 1623, 8937, 50565, 292283, 1718827, 10250916, 61854848, 376949934, 2316738789, 14343701657, 89379109846, 560108223900, 3527723269978, 22318890516413, 141778326349191, 903936594232782, 5782447430948438, 37102633354583532, 238729798670985104
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 61*x^4 + 306*x^5 + 1623*x^6 +...
where the logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A)*x*A + (1 + 2^3*x*A + x^2*A^2)*x^2*A^2/2 +
(1 + 3^3*x*A + 3^3*x^2*A^2 + x^3*A^3)*x^3*A^3/3 +
(1 + 4^3*x*A + 6^3*x^2*A^2 + 4^3*x^3*A^3 + x^4*A^4)*x^4*A^4/4 +
(1 + 5^3*x*A + 10^3*x^2*A^2 + 10^3*x^3*A^3 + 5^3*x^4*A^4 + x^5*A^5)*x^5*A^5/5 +
(1 + 6^3*x*A + 15^3*x^2*A^2 + 20^3*x^3*A^3 + 15^3*x^4*A^4 + 6^3*x^5*A^5 + x^6*A^6)*x^6*A^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 31*x^3/3 + 185*x^4/4 + 1126*x^5/5 + 7043*x^6/6 + 44689*x^7/7 + 286241*x^8/8 +...

Crossrefs

Cf. A198944, A200212, A192131, A166896, A199875, A166990, A198950, A181143, A181543.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*x^j*A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}
Showing 1-5 of 5 results.

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