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A091597
Triangle read by rows: T(n,0) = A001045(n+1), T(n,n)=1, T(n,m) = T(n-1,m-1) + T(n-1,m).
2
1, 1, 1, 3, 2, 1, 5, 5, 3, 1, 11, 10, 8, 4, 1, 21, 21, 18, 12, 5, 1, 43, 42, 39, 30, 17, 6, 1, 85, 85, 81, 69, 47, 23, 7, 1, 171, 170, 166, 150, 116, 70, 30, 8, 1, 341, 341, 336, 316, 266, 186, 100, 38, 9, 1, 683, 682, 677, 652, 582, 452, 286, 138, 47, 10, 1
OFFSET
0,4
COMMENTS
A Jacobsthal-Pascal triangle.
Equals triangle M * Pascal's triangle, M = an infinite lower triangular Toeplitz matrix with A078008: [1, 0, 2, 2, 6, 10, 22, 42, ...] in every column. - Gary W. Adamson, May 25 2009
FORMULA
Number triangle: T(n, k) = Sum_{j=0..n} binomial(n-j, k+j)2^j.
Riordan array: (1/(1-x-2*x^2), x/(1-x)).
k-th column has g.f. (1/(1-x-2*x^2))*(x/(1-x))^k.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - 2*T(n-3,k) - 2*T(n-3,k-1), T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=3, T(2,1)=2, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 11 2014
EXAMPLE
Triangle begins as:
1;
1, 1;
3, 2, 1;
5, 5, 3, 1;
11, 10, 8, 4, 1;
21, 21, 18, 12, 5, 1;
43, 42, 39, 30, 17, 6, 1;
85, 85, 81, 69, 47, 23, 7, 1;
171, 170, 166, 150, 116, 70, 30, 8, 1;
341, 341, 336, 316, 266, 186, 100, 38, 9, 1;
MAPLE
A091597 := proc(n, k)
if k = 0 then
A001045(n+1) ;
elif k = n then
1 ;
elif k <0 or k > n then
0 ;
else
procname(n-1, k-1)+procname(n-1, k) ;
end if;
end proc: # R. J. Mathar, Oct 05 2012
MATHEMATICA
Table[Sum[Binomial[n-j, k+j]*2^j, {j, 0, n}], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 04 2019 *)
PROG
(PARI) {T(n, k) = sum(j=0, n, 2^j*binomial(n-j, k+j))}; \\ G. C. Greubel, Jun 04 2019
(Magma) [[(&+[2^j*Binomial(n-j, k+j): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 04 2019
(SageMath) [[sum(2^j*binomial(n-j, k+j) for j in (0..n)) for k in (0..n)] for n in [0..12]] # G. C. Greubel, Jun 04 2019
(GAP) Flat(List([0..12], n->List([0..n], k-> Sum([0..n], j-> 2^j*Binomial(n-j, k+j)) ))); # G. C. Greubel, Jun 04 2019
CROSSREFS
Columns include A001045, A000975, A011377.
Row sums are A059570.
Cf. A078008. - Gary W. Adamson, May 25 2009
Sequence in context: A021315 A337315 A068389 * A091595 A246837 A132969
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Jan 23 2004
STATUS
approved