A102400
Triangle, read by rows, where T(n,k) = Sum_{j=0..k} T(n-1,j)*(j+1)*[(k+1)*(k+2)/2 - j*(j+1)/2] for n>k>0, with T(0,0)=1 and T(n,n) = T(n,n-1) for n>0.
Original entry on oeis.org
1, 1, 1, 1, 7, 7, 1, 31, 139, 139, 1, 127, 1567, 5711, 5711, 1, 511, 15379, 126579, 408354, 408354, 1, 2047, 143527, 2357431, 15333661, 45605881, 45605881, 1, 8191, 1312219, 40769819, 473433344, 2634441290, 7390305396, 7390305396, 1, 32767
Offset: 0
T(4,2) = 1567 = 1*6 + 31*10 + 139*9
= T(3,0)*R(0,2) + T(3,1)*R(1,2) + T(3,2)*R(2,2).
Rows begin:
[1],
[1,1],
[1,7,7],
[1,31,139,139],
[1,127,1567,5711,5711],
[1,511,15379,126579,408354,408354],
[1,2047,143527,2357431,15333661,45605881,45605881],...
where the transpose of the recurrence coefficients given by
[R^t](n,k) = (k+1)*((n+1)*(n+2)/2 - k*(k+1)/2) form triangle:
[1],
[3,4],
[6,10,9],
[10,18,21,16],
[15,28,36,36,25],...
which equals the matrix square of the triangle:
[1],
[1,2],
[1,2,3],
[1,2,3,4],
[1,2,3,4,5],...
-
T[n_, k_] := T[n, k] = If[nJean-François Alcover, Dec 15 2014, after PARI *)
-
{T(n,k)=if(n
Original entry on oeis.org
0, 2, 4, 8, 40, 152, 1128, 6200, 61120, 442552, 5466320, 49399320, 735847800, 8003532512, 139910204080, 1784040237288, 35858685086352, 525504809786112, 11953187179149408, 198213959637435608, 5037776918810353960
Offset: 0
2 = 2*(1-2x) + 4*x*(1-2x) + 8*x^2*(1-2x)(1-3x) + 40*x^3*(1-2x)(1-3x)
+ 152*x^4*(1-2x)(1-3x)(1-4x) + 1128*x^5*(1-2x)(1-3x)(1-4x)
+ 6200*x^6*(1-2x)(1-3x)(1-4x)(1-5x) + 61120*x^7*(1-2x)(1-3x)(1-4x)(1-5x) +...
+ A102099(n+1)*x^(2n)*(1-2x)(1-3x)*..*(1-(n+2)x)
+ A102922(n+1)*x^(2n+1)*(1-x)(1-2x)*..*(1-(n+2)x) + ...
-
{a(n)=if(n==0,2,polcoeff(2-sum(k=0,n-1,a(k)*x^k*prod(j=2,k\2+2,1-j*x+x*O(x^n))),n))}
A103241
Unreduced numerators of the elements T(n,k)/(n-k)!, read by rows, of the triangular matrix P^-1, which is the inverse of the matrix defined by P(n,k) = (-k^3)^(n-k)/(n-k)! for n >= k >= 1.
Original entry on oeis.org
1, 1, 1, 15, 8, 1, 1024, 368, 27, 1, 198581, 53672, 2727, 64, 1, 85102056, 18417792, 710532, 11904, 125, 1, 68999174203, 12448430408, 386023509, 4975936, 38375, 216, 1, 95264160938080, 14734002979456, 381535651512, 3977848832, 23945000
Offset: 1
Rows of unreduced fractions T(n,k)/(n-k)! begin:
[1/0!],
[1/1!, 1/0!],
[15/2!, 8/1!, 1/0!],
[1024/3!, 368/2!, 27/1!, 1/0!],
[198581/4!, 53672/3!, 2727/2!, 64/1!, 1/0!],
[85102056/5!, 18417792/4!, 710532/3!, 11904/2!, 125/1!, 1/0!], ...
forming the inverse of matrix P where P(n,k) = A103246(n,k)/(n-k)!:
[1/0!],
[-1/1!, 1/0!],
[1/2!, -8/1!, 1/0!],
[-1/3!, 64/2!, -27/1!, 1/0!],
[1/4!, -512/3!, 729/2!, -64/1!, 1/0!], ...
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{T(n,k)=my(P);if(n>=k&k>=1, P=matrix(n,n,r,c,if(r>=c,(-c^3)^(r-c)/(r-c)!))); return(if(n
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