A102223
Column 0 of triangular matrix A102222, which equals -log[2*I - A008459].
Original entry on oeis.org
0, 1, 3, 22, 323, 7906, 290262, 14919430, 1022475715, 90094491994, 9923239949978, 1335853771297750, 215797095378591542, 41198645313603207990, 9176288655853717238830, 2358300288047799986966722
Offset: 0
a(2) = 3 = 1 + (1*0*0 + 4*1*1)/2,
a(3) = 22 = 1 + (1*0*0 + 9*1*1 + 9*2*3)/3,
a(4) = 323 = 1 + (1*0*0 + 16*1*1 + 36*2*3 + 16*3*22)/4,
a(5) = 7906 = 1 + (1*0*0 + 25*1*1 + 100*2*3 + 100*3*22 + 25*4*323)/5.
-
a(n)=if(n<1,0,1+sum(k=0,n-1,binomial(n,k)^2*k*a(k))/n)
A102221
Column 0 of triangular matrix A102220, which equals [2*I - A008459]^(-1).
Original entry on oeis.org
1, 1, 5, 55, 1077, 32951, 1451723, 87054773, 6818444405, 675900963271, 82717196780955, 12248810651651333, 2158585005685222491, 446445657799551807541, 107087164031952038620481, 29487141797206760561836055, 9238158011747884080353808245
Offset: 0
-
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)*binomial(n, i)/i!, i=1..n))
end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20); # Alois P. Heinz, May 11 2016
-
Rest[CoefficientList[Series[1/(2-BesselJ[0, 2*I*Sqrt[x]]), {x, 0, 20}], x] * Range[0, 20]!^2] (* Vaclav Kotesovec, Mar 02 2014 *)
m = 20; CoefficientList[1/(2 - BesselI[0, 2 Sqrt[x]]) + O[x]^m, x] Range[0, m - 1]!^2 (* Jean-François Alcover, Jun 11 2019, after Vladeta Jovovic *)
b[n_] := b[n] = If[n==0, 1, Sum[b[n-i] Binomial[n, i]/i!, {i, 1, n}]];
a[n_] := b[n] n!;
a /@ Range[0, 20] (* Jean-François Alcover, Dec 03 2020, after Alois P. Heinz *)
-
a(n)=if(n==0,1,sum(k=0,n-1,binomial(n,k)^2*a(k)))
-
L = taylor(1/(1-x*hypergeometric((1,),(2,2),x)),x,0,14).list()
[factorial(i)^2*c for (i,c) in enumerate(L)] # Peter Luschny, Jul 28 2015
A102220
Triangular matrix, read by rows, equal to [2*I - A008459]^(-1), i.e., the matrix inverse of the difference of twice the identity matrix and the triangular matrix of squared binomial coefficients.
Original entry on oeis.org
1, 1, 1, 5, 4, 1, 55, 45, 9, 1, 1077, 880, 180, 16, 1, 32951, 26925, 5500, 500, 25, 1, 1451723, 1186236, 242325, 22000, 1125, 36, 1, 87054773, 71134427, 14531391, 1319325, 67375, 2205, 49, 1, 6818444405, 5571505472, 1138150832, 103334336, 5277300, 172480, 3920, 64, 1
Offset: 0
Rows begin:
[1],
[1,1],
[5,4,1],
[55,45,9,1],
[1077,880,180,16,1],
[32951,26925,5500,500,25,1],
[1451723,1186236,242325,22000,1125,36,1],...
and equal the term-by-term product of column 0
with the squared binomial coefficients (A008459):
[(1)1^2],
[(1)1^2,(1)1^2],
[(5)1^2,(1)2^2,(1)1^2],
[(55)1^2,(5)3^2,(1)3^2,(1)1^2],
[(1077)1^2,(55)4^2,(5)6^2,(1)4^2,(1)1^2],...
The matrix inverse is [2*I - A008459]:
[1],
[ -1,1],
[ -1,-4,1],
[ -1,-9,-9,1],
[ -1,-16,-36,-16,1],...
-
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)*binomial(n, i)/i!, i=1..n))
end:
T:= (n, k)-> binomial(n, k)^2*b(n-k)*(n-k)!:
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Sep 10 2019
-
nmax = 10;
M = Inverse[2 IdentityMatrix[nmax+1] - Table[Binomial[n, k]^2, {n, 0, nmax}, {k, 0, nmax}]];
T[n_, k_] := M[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)
-
{T(n,k)=(matrix(n+1,n+1,i,j,if(i==j,2,0)-binomial(i-1,j-1)^2)^-1)[n+1,k+1]}
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