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.

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A124280 Expansion of 1/(1-x-x^2+x^3-x^4).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 9, 12, 20, 28, 45, 65, 102, 150, 232, 345, 529, 792, 1208, 1816, 2761, 4161, 6314, 9530, 14444, 21821, 33049, 49956, 75628, 114356, 173077, 261761, 396110, 599150, 906576, 1371377, 2074913, 3138864, 4748976, 7184304, 10869329, 16443521
Offset: 0

Views

Author

Paul Barry, Oct 24 2006

Keywords

Comments

Diagonal sums of number triangle A124279.

Links

Crossrefs

Cf. A124279.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x-x^2+x^3-x^4),{x,0,37}],x] (* Stefano Spezia, Aug 25 2025 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2k}, C(j,n-2k-j)*C(k,n-2k-j) for n > 0.
a(n) = a(n-1)+a(n-2)-a(n-3)+a(n-4) for n > 3. - Carl Najafi, May 06 2014

A124445 Expansion of 1/(1-x-x*y+x^2*y-x^3*y^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 5, 4, 1, 1, 1, 5, 7, 8, 5, 1, 1, 1, 6, 9, 13, 12, 6, 1, 1, 1, 7, 11, 19, 22, 17, 7, 1, 1, 1, 8, 13, 26, 35, 35, 23, 8, 1, 1, 1, 9, 15, 34, 51, 61, 53, 30, 9, 1
Offset: 0

Views

Author

Paul Barry, Nov 01 2006

Keywords

Comments

Row sums are A107293(n+4). Diagonal sums are A005314(n+1).
Reversal of A124279. T(2n,n) is A002426. - Paul Barry, Nov 01 2006

Examples

			Triangle begins:
1,
1, 1,
1, 1, 1,
1, 1, 2, 1,
1, 1, 3, 3, 1,
1, 1, 4, 5, 4, 1,
1, 1, 5, 7, 8, 5, 1,
1, 1, 6, 9, 13, 12, 6, 1,
1, 1, 7, 11, 19, 22, 17, 7, 1

Crossrefs

Cf. A180562.

Formula

Number triangle T(n,k)=sum{j=0..n, C(j,k-j)*C(n-k,k-j)}*[k<=n];
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + T(n-3,k-2), T(0,0) = T(1,0) = T(1,1) = T(2,0) = T(2,1) = T(2,2) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2014
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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