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.

A185755 Triangle: T(n,k) equals the coefficient of x^n*y^k in the n-th iteration of x*(1+xy)/(1-x), for n>=1, 0<=k

Original entry on oeis.org

1, 2, 2, 9, 15, 6, 64, 154, 120, 30, 625, 1995, 2340, 1190, 220, 7776, 31191, 49315, 38325, 14595, 2170, 117649, 571221, 1142932, 1204588, 704102, 215950, 27076, 2097152, 11992688, 29141994, 38972388, 30945432, 14570976, 3761310, 409836
Offset: 1

Views

Author

Paul D. Hanna, Feb 03 2011

Keywords

Examples

			Triangle begins:
1;
2, 2;
9, 15, 6;
64, 154, 120, 30;
625, 1995, 2340, 1190, 220;
7776, 31191, 49315, 38325, 14595, 2170;
117649, 571221, 1142932, 1204588, 704102, 215950, 27076;
2097152, 11992688, 29141994, 38972388, 30945432, 14570976, 3761310, 409836;
43046721, 283976517, 814059798, 1323693384, 1334427720, 853356072, 337738758, 75550188, 7303164; ...

Crossrefs

Cf. columns: A000169, A185756, A185757; row sums: A185523.
Cf. diagonals: A112317, A185758, A185759.

Programs

  • PARI
    {T(n,k)=local(A=x, G=x*(1+x*y)/(1-x)); for(i=1, n, A=subst(G, x, A+x*O(x^n)));polcoeff(polcoeff(A, n,x),k,y)}

Formula

T(n,0) = A000169(n) = n^(n-1).
T(n,n) = A112317(n).
Sum_{k=0..n-1} T(n,k) = A185523(n).
Sum_{k=0..n-1} (-1)^k*T(n,k) = 0^n.

A185522 a(n) equals the coefficient of x^n in the (n-1)-th iteration of x*(1+x)/(1-x) for n>=1.

Original entry on oeis.org

1, 2, 12, 138, 2320, 51450, 1418004, 46736466, 1793145792, 78506270994, 3862498271324, 210975923301242, 12668208032568400, 829409050807729002, 58804315058897866020, 4488388292080413635362, 366956820758560590789376
Offset: 1

Views

Author

Paul D. Hanna, Jan 30 2011

Keywords

Examples

			 Given G(x) = x*(1+x)/(1-x):
G(x) = x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 +...
then the initial coefficients of the n-th iterations of G(x) begin:
n=0: [(1), 0, 0, 0, 0, 0, 0, ...];
n=1: [1,(2), 2, 2, 2, 2, 2, 2, 2, ...];
n=2: [1, 4,(12), 32, 80, 196, 476, 1152, 2784, ...];
n=3: [1, 6, 30,(138), 602, 2542, 10518, 42994, ...];
n=4: [1, 8, 56, 368,(2320), 14216, 85368, 505312, ...];
n=5: [1, 10, 90, 770, 6370,(51450), 408202, 3194978, ...];
n=6: [1, 12, 132, 1392, 14272, 143372,(1418004), 13854368, ...];
n=7: [1, 14, 182, 2282, 27930, 335846, 3983518,(46736466), ...]; ...;
the coefficients in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A185523, A185524.

A185524 a(n) equals the coefficient of x^n in the (n+1)-th iteration of x*(1+x)/(1-x) for n>=1.

Original entry on oeis.org

1, 6, 56, 770, 14272, 335846, 9623280, 325812162, 12743851808, 565954103110, 28147009533480, 1550288951887650, 93697417382512608, 6166356881177224390, 439006462312153564128, 33620884878446290152706, 2756259421284677015952320
Offset: 1

Views

Author

Paul D. Hanna, Jan 30 2011

Keywords

Examples

			 Given G(x) = x*(1+x)/(1-x):
G(x) = x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 +...
then the initial coefficients of the n-th iterations of G(x) begin:
n=1: [1, 2, 2, 2, 2, 2, 2, 2, 2, ...];
n=2: [(1), 4, 12, 32, 80, 196, 476, 1152, 2784, ...];
n=3: [1,(6), 30, 138, 602, 2542, 10518, 42994, ...];
n=4: [1, 8,(56), 368, 2320, 14216, 85368, 505312, ...];
n=5: [1, 10, 90,(770), 6370, 51450, 408202, 3194978, ...];
n=6: [1, 12, 132, 1392,(14272), 143372, 1418004, 13854368, ...];
n=7: [1, 14, 182, 2282, 27930,(335846), 3983518, 46736466, ...];
n=8: [1, 16, 240, 3488, 49632, 695312,(9623280), 131891776, ...];
n=9: [1, 18, 306, 5058, 82050, 1312626, 20771730,(325812162), ...]; ...;
the coefficients in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A185522, A185523.
Showing 1-3 of 3 results.

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