A117396 Triangle, read by rows, defined by: T(n,k) = (k+1)*T(n,k+1) - Sum_{j=1..n-k-1} T(j,0)*T(n,j+k+1) for n>k with T(n,n)=1 for n>=0.
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 11, 4, 1, 1, 77, 51, 19, 5, 1, 1, 437, 291, 109, 29, 6, 1, 1, 2957, 1971, 739, 197, 41, 7, 1, 1, 23117, 15411, 5779, 1541, 321, 55, 8, 1, 1, 204557, 136371, 51139, 13637, 2841, 487, 71, 9, 1, 1, 2018957, 1345971, 504739, 134597, 28041, 4807, 701, 89, 10, 1
Offset: 0
Examples
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 5, 3, 1;
1, 17, 11, 4, 1;
1, 77, 51, 19, 5, 1;
1, 437, 291, 109, 29, 6, 1;
1, 2957, 1971, 739, 197, 41, 7, 1;
1, 23117, 15411, 5779, 1541, 321, 55, 8, 1;
1, 204557, 136371, 51139, 13637, 2841, 487, 71, 9, 1; ...
Matrix inverse is:
1;
-1, 1;
1, -2, 1;
1, 1, -3, 1;
1, 1, 1, -4, 1;
1, 1, 1, 1, -5, 1; ...
Matrix log is the integer triangle A117398:
0;
1, 0;
0, 2, 0;
-1, 2, 3, 0;
-3, 4, 5, 4, 0;
-9, 14, 15, 9, 5, 0;
-33, 68, 65, 34, 14, 6, 0; ...
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
-
Magma
[k eq 0 select 1 else k*(&+[Factorial(j)/Factorial(k+1): j in [k-1..n]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 24 2021
-
Mathematica
T[n_, k_]:= T[n, k]= If[k==0, 1, k*Sum[j!/(k+1)!, {j,k-1,n}]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 24 2021 *) -
PARI
T(n,k)=if(n
-
PARI
/* Definition by Matrix Inverse: */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,if(r==c+1,-c,1))));(M^-1)[n+1,k+1]
-
PARI
T(n,k)=if(n
Paul D. Hanna, Jun 20 2006 -
Sage
def A117396(n,k): return 1 if (k==0) else k*sum(factorial(j)/factorial(k+1) for j in (k-1..n)) flatten([[A117396(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 24 2021
Formula
T(n,k) = k*Sum_{j=k-1..n} j!/(k+1)! for n >= k > 0, with T(n,0) = 1 for n >= 0. - Paul D. Hanna, Jun 20 2006
Comments