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A192915
Triangle read by rows: T(n,k) = Sum_{j=0..3} binomial(n+3, k+j), 0 <= k <= n.
2
8, 15, 15, 26, 30, 26, 42, 56, 56, 42, 64, 98, 112, 98, 64, 93, 162, 210, 210, 162, 93, 130, 255, 372, 420, 372, 255, 130, 176, 385, 627, 792, 792, 627, 385, 176, 232, 561, 1012, 1419, 1584, 1419, 1012, 561, 232, 299, 793, 1573, 2431, 3003, 3003, 2431
OFFSET
0,1
FORMULA
T(n,0) = T(n,n) = A000125(n+3); T(n,k) = T(n-1,k-1) + T(n-1,k) for 0 < k < n. - Georg Fischer, Nov 26 2021
T(n, k) = binomial(n + 4, k + 1) + binomial(n + 4, k + 3). - Peter Luschny, Nov 26 2021
EXAMPLE
Northwest corner:
8;
15, 15;
26, 30, 26;
42, 56, 56, 42;
64, 98, 112, 98, 64;
MAPLE
A192915 := proc(n, k) binomial(n+3, k) +binomial(n+3, k+1) +binomial(n+3, k+2) +binomial(n+3, k+3) ; end proc: # R. J. Mathar, Aug 25 2011
MATHEMATICA
T[n_, k_]:= Sum[Binomial[n+3, k+j], {j, 0, 3}]
Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* A192915 as a sequence *)
TableForm[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* A192915 as an array *)
PROG
(PARI) for(n=0, 10, for(k=0, n, print1(sum(j=0, 3, binomial(n+3, k+j)), ", "))) \\ G. C. Greubel, Jan 12 2019
(Magma) [[(&+[Binomial(n+3, k+j): j in [0..3]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 12 2019
(Sage) [[sum(binomial(n+3, k+j) for j in (0..3)) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Jan 12 2019
(GAP) T:=Flat(List([0..10], n->List([0..n], k->Sum([0..3], j-> Binomial(n+3, k+j) )))); # G. C. Greubel, Jan 12 2019
CROSSREFS
Sequence in context: A126852 A248389 A301618 * A229839 A114605 A300860
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 16 2011
STATUS
approved