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.

A057995 Coefficient triangle of polynomials (rising powers) related to Fibonacci convolutions. Companion triangle to A057280.

Original entry on oeis.org

1, 16, 5, 300, 160, 20, 6840, 4850, 1075, 75, 186120, 159650, 48175, 6100, 275, 5916240, 5846700, 2168650, 379700, 31550, 1000, 215717040, 238437900, 103057800, 22426825, 2605175, 153875, 3625, 8888140800, 10772348400
Offset: 0

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Author

Wolfdieter Lang, Sep 13 2000

Keywords

Comments

The row polynomials are p(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...
The k-th convolution of F0(n) := A000045(n+1) n >= 0, (Fibonacci starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) = (p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k)), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A057280).

Examples

			k=2: F2(n)=((16+5*n)*(n+1)*F0(n+1)+(17+5*n)*(n+2)*F0(n))/50, cf. A001628.

Crossrefs

Cf. A000045, A037027, A057280.

A057282 Coefficient triangle of polynomials (falling powers) related to Fibonacci convolutions. Companion triangle to A057281.

Original entry on oeis.org

2, 5, 17, 15, 120, 225, 50, 700, 3050, 4080, 175, 3775, 28625, 89225, 94440, 625, 19225, 223175, 1208975, 3006000, 2666880, 2250, 93500, 1537100, 12689800, 54824650, 115299900, 89016480, 8125, 438250, 9670750, 112454500, 737744125
Offset: 1

Views

Author

Wolfdieter Lang, Sep 13 2000

Keywords

Comments

The row polynomials are q(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..
The k-th convolution of F0(n) := A000045(n+1), n >= 0, (Fibonacci numbers starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) = (p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A057281(k,m).
a(k,0)= A020876(k), k >= 0.

Examples

			k=2: F2(n)=((5*n^2+21*n+16)*F(n+2)+(5*n^2+27*n+34)*F(n+1))/50, F(n) := A000045(n); see A001628.
2; 5,17; 15,120,225; 50,700,3050,4080; 175,3775,28625,89225,94440; ...

Crossrefs

Cf. A000045, A037027, A057995, A057280, A057281.
Row sums: A151615.

A057281 Coefficient triangle of polynomials (falling powers) related to Fibonacci convolutions. Companion triangle to A057282.

Original entry on oeis.org

1, 5, 16, 20, 160, 300, 75, 1075, 4850, 6840, 275, 6100, 48175, 159650, 186120, 1000, 31550, 379700, 2168650, 5846700, 5916240, 3625, 153875, 2605175, 22426825, 103057800, 238437900, 215717040, 13125, 720375, 16273875, 195469125
Offset: 0

Views

Author

Wolfdieter Lang, Sep 13 2000

Keywords

Comments

The row polynomials are p(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..
The k-th convolution of F0(n) := A000045(n+1), n >= 0, (Fibonacci numbers starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) = (p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^(k-m),m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A057282(k,m).
a(k,0)= A030191(k), k >= 0.

Examples

			k=2: F2(n)=((5*n^2+21*n+16)*F(n+2)+(5*n^2+27*n+34)*F(n+1))/50, F(n)=A000045(n); see A001628.

Crossrefs

Cf. A000045, A037027, A057995, A057280, A057282.
Showing 1-3 of 3 results.

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