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 1-4 of 4 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

Views

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(""))

A181144 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*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, 9, 1, 1, 36, 36, 1, 1, 100, 419, 100, 1, 1, 225, 2699, 2699, 225, 1, 1, 441, 12138, 35052, 12138, 441, 1, 1, 784, 42865, 286206, 286206, 42865, 784, 1, 1, 1296, 127191, 1696820, 3932898, 1696820, 127191, 1296, 1, 1, 2025, 330903, 7958563
Offset: 0

Views

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 );
* A181143: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*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+9*y+y^2)*x^2 + (1+36*y+36*y^2+y^3)*x^3 + (1+100*y+419*y^2+100*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^4*y + y^2)*x^2/2
+ (1 + 3^4*y + 3^4*y^2 + y^3)*x^3/3
+ (1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4)*x^4/4
+ (1 + 5^4*y + 10^4*y^2 + 10^4*y^3 + 5^4*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 9, 1;
1, 36, 36, 1;
1, 100, 419, 100, 1;
1, 225, 2699, 2699, 225, 1;
1, 441, 12138, 35052, 12138, 441, 1;
1, 784, 42865, 286206, 286206, 42865, 784, 1;
1, 1296, 127191, 1696820, 3932898, 1696820, 127191, 1296, 1;
1, 2025, 330903, 7958563, 36955542, 36955542, 7958563, 330903, 2025, 1;
1, 3025, 776688, 31205941, 261852055, 525079969, 261852055, 31205941, 776688, 3025, 1; ...
Note that column 1 forms the sum of cubes (A000537), and forms the squares of the triangular numbers.
Inverse binomial transform of columns begins:
[1];
[1, 8, 19, 18, 6];
[1, 35, 348, 1549, 3713, 5154, 4161, 1818, 333];
[1, 99, 2500, 27254, 161793, 589819, 1409579, 2282850, 2529900, 1893972, 917349, 259854, 32726]; ...

Crossrefs

Cf. A000537 (column 1), A166992 (row sums), A166898 (antidiagonal sums), A218140.
Cf. variants: A001263 (Narayana), A181143, A218115, A218116.

Programs

  • PARI
    {T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,sum(j=0,m,binomial(m,j)^4*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(""))

A218115 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5 * 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, 17, 1, 1, 98, 98, 1, 1, 354, 2251, 354, 1, 1, 979, 23803, 23803, 979, 1, 1, 2275, 158367, 617036, 158367, 2275, 1, 1, 4676, 773842, 8763293, 8763293, 773842, 4676, 1, 1, 8772, 3031668, 82498785, 241082026, 82498785, 3031668, 8772, 1, 1, 15333
Offset: 0

Views

Author

Paul D. Hanna, Oct 21 2012

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 );
* A181143: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*y^k] * x^n/n );
* A181144: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*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+17*y+y^2)*x^2 + (1+98*y+98*y^2+y^3)*x^3 + (1+354*y+2251*y^2+354*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^5*y + y^2)*x^2/2
+ (1 + 3^5*y + 3^5*y^2 + y^3)*x^3/3
+ (1 + 4^5*y + 6^5*y^2 + 4^5*y^3 + y^4)*x^4/4
+ (1 + 5^5*y + 10^5*y^2 + 10^5*y^3 + 5^5*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 17, 1;
1, 98, 98, 1;
1, 354, 2251, 354, 1;
1, 979, 23803, 23803, 979, 1;
1, 2275, 158367, 617036, 158367, 2275, 1;
1, 4676, 773842, 8763293, 8763293, 773842, 4676, 1;
1, 8772, 3031668, 82498785, 241082026, 82498785, 3031668, 8772, 1;
1, 15333, 10057620, 575963523, 4066874561, 4066874561, 575963523, 10057620, 15333, 1; ...
Note that column 1 forms the sum of fourth powers (A000538).

Crossrefs

Cf. A000538 (column 1), A218117 (row sums).
Cf. variants: A001263 (Narayana), A181143, A181144, A218116.

Programs

  • PARI
    {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^5*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(""))

A218119 G.f.: A(x) = exp( Sum_{n>=1} A069865(n)*x^n/n ) where A069865(n) = Sum_{k=0..n} C(n,k)^6.

Original entry on oeis.org

1, 2, 35, 554, 15297, 451842, 15929824, 601077640, 24488754772, 1046792248856, 46718718597567, 2155032002133834, 102259392504591235, 4967499746642163574, 246231868462969357492, 12419324761881256326288, 635990044563649443993091, 33006906229799699591298070
Offset: 0

Views

Author

Paul D. Hanna, Oct 21 2012

Keywords

Comments

Compare to a g.f. of Catalan numbers (A000108):
exp( Sum_{n>=1} A000984(n)*x^n/n ) where A000984(n) = Sum_{k=0..n} C(n,k)^2.

Examples

			G.f.: A(x) = 1 + 2*x + 35*x^2 + 554*x^3 + 15297*x^4 + 451842*x^5 + 15929824*x^6 +...
log(A(x)) = 2*x + 66*x^2/2 + 1460*x^3/3 + 54850*x^4/4 + 2031252*x^5/5 + 86874564*x^6/6 + 3848298792*x^7/7 +...+ A069865(n)*x^n/n +...

Crossrefs

Cf. A218116, A218120, A166990, A166992, A218117, A069865.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^6)*x^m/m)+x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))

Formula

Equals row sums of triangle A218116.
Self-convolution of A218120.
Showing 1-4 of 4 results.

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