A098518 -id:A098518 - cp's OEIS frontend

A098521 E.g.f. exp(x)BesselI(2,2sqrt(2)*x)/2.

0, 0, 1, 3, 14, 50, 195, 721, 2716, 10116, 37845, 141295, 528330, 1975766, 7395479, 27698685, 103821240, 389410568, 1461605481, 5489516955, 20630539910, 77579118330, 291893775019, 1098848179561, 4138773239892, 15596070165900, 58797332264125, 221762856917511, 836756771788098

Offset: 0

Paul Barry, Sep 12 2004

Binomial transform of e.g.f. BesselI(2,2*sqrt(2)*x)/2, or {0,0,1,0,8,0,60,0,448,0,3360,...} with g.f. ((1-4*x^2)-sqrt(1-8*x^2))/(8*x^2*sqrt(1-8*x^2)).

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A014531, A098518.

m:=40; R:=PowerSeriesRing(Rationals(), m); [0,0] cat Coefficients(R!((1-2*x-3*x^2-(1-x)*Sqrt(1-2*x-7*x^2))/(8*x^2*Sqrt(1-2*x-7*x^2)))); // G. C. Greubel, Aug 17 2018

CoefficientList[Series[(1-2*x-3*x^2-(1-x)*Sqrt[1-2*x-7*x^2]) / (8*x^2*Sqrt[1-2*x-7*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 28 2012 *)

x='x+O('x^66); concat([0,0],Vec((1-2*x-3*x^2-(1-x)*sqrt(1-2*x-7*x^2))/(8*x^2*sqrt(1-2*x-7*x^2)))) \\ Joerg Arndt, May 11 2013

G.f.: (1-2*x-3*x^2-(1-x)*sqrt(1-2*x-7*x^2))/(8*x^2*sqrt(1-2*x-7*x^2)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k+2)*2^k.
Conjecture: (n+2)*a(n) -(4n+3)*a(n-1) -3*(2n+1)*a(n-2) +(20n-29)*a(n-3) +21*(n-3)*a(n-4)=0. - R. J. Mathar, Dec 08 2011
Shorter recurrence (for n>=3): (n-2)*(n+2)*a(n) = n*(2*n-1)*a(n-1) + 7*(n-1)*n*a(n-2). - Vaclav Kotesovec, Dec 28 2012
a(n) ~ sqrt(8+2*sqrt(2))*(1+2*sqrt(2))^n/(8*sqrt(Pi*n)). - Vaclav Kotesovec, Dec 28 2012

A115991 Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)C(j,n-j)2^(n-j).

Original entry on oeis.org

1, 1, 1, 5, 3, 1, 13, 9, 5, 1, 49, 31, 17, 7, 1, 161, 105, 61, 29, 9, 1, 581, 371, 217, 111, 45, 11, 1, 2045, 1313, 781, 417, 189, 65, 13, 1, 7393, 4719, 2825, 1551, 753, 303, 89, 15, 1, 26689, 17041, 10277, 5757, 2921, 1289, 461, 117, 17, 1

Offset: 0

Paul Barry, Feb 10 2006

First column is A084601 with e.g.f. exp(x) Bessel_I(0,2*sqrt(2)x). Row sums are A098518(n+1) with e.g.f. dif(exp(x) Bessel_I(1,2*sqrt(2)x)/sqrt(2)).

Riordan array (1/sqrt(1-2*x-7*x^2), (1+x-sqrt(1-2*x-7*x^2))/2).

			Triangle begins as:
    1;
    1,   1;
    5,   3,   1;
   13,   9,   5,   1;
   49,  31,  17,   7,  1;
  161, 105,  61,  29,  9,  1;
  581, 371, 217, 111, 45, 11, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A084601 (k=0), A098518.

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-j)*2^(n-j)) ))); # G. C. Greubel, May 09 2019

[[(&+[Binomial(n-k, j-k)*Binomial(j, n-j)*2^(n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019

A115991 := proc(n,k)
    add(binomial(n-k,j-k)*binomial(j,n-j)*2^(n-j),j=0..n) ;
end proc:
seq(seq(A115991(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jun 25 2023

Table[Sum[Binomial[n-k, j-k]*Binomial[j, n-j]*2^(n-j), {j, 0, n}], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 09 2019 *)

{T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j)*2^(n-j))}; \\ G. C. Greubel, May 09 2019

[[sum(binomial(n-k, j-k)*binomial(j, n-j)*2^(n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019

cp's OEIS Frontend

A098521 E.g.f. exp(x)BesselI(2,2sqrt(2)*x)/2.

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A115991 Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)C(j,n-j)2^(n-j).

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cp's OEIS Frontend

A098521 E.g.f. exp(x)*BesselI(2,2*sqrt(2)*x)/2.

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Magma

Mathematica

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Formula

A115991 Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j)*2^(n-j).

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GAP

Magma

Maple

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Sage

A098521 E.g.f. exp(x)BesselI(2,2sqrt(2)*x)/2.

A115991 Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)C(j,n-j)2^(n-j).