A180958 Diagonal sums of generalized Narayana triangle A180957.

1, 1, 2, 2, 2, -1, -8, -25, -57, -114, -202, -322, -447, -496, -271, 625, 2914, 7762, 16834, 32063, 54760, 83319, 108375, 103726, 11110, -282498, -973439, -2366432, -4869919, -8903455, -14604094, -21135454, -25294718, -19009153, 14697432, 107405319, 311830247, 705982670, 1386882198, 2436851006, 3830805953

Offset: 0

Author

Paul Barry, Sep 28 2010

Keywords

• easy
• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1).

Crossrefs

Cf. A180957.

Programs

Magma

I:=[1,1,2,2]; [n le 4 select I[n] else 2*Self(n-1) +Self(n-2) -3*Self(n-3) -Self(n-4): n in [1..51]]; // G. C. Greubel, Apr 06 2021

Mathematica

LinearRecurrence[{2,1,-3,-1}, {1,1,2,2}, 51] (* G. C. Greubel, Apr 06 2021 *)

Sage

[sum( sum( (-1)^(k-j)*binomial(n-k, j)*binomial(n-k-j, 2*(k-j)) for j in (0..n-k)) for k in (0..n//2)) for n in (0..50)] # G. C. Greubel, Apr 06 2021

Formula

G.f.: ( 1-x-x^2 ) / ( (1+x)*(1-3*x+2*x^2+x^3) ).

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} (-1)^(k-j) * binomial(n-k, j) * binomial(n-k-j, 2*(k-j)).

Extensions

Terms a(31) onward added by G. C. Greubel, Apr 06 2021