A124051 Quasi-mirror of A062196 formatted as a triangular array.

3, 6, 8, 10, 30, 15, 15, 80, 90, 24, 21, 175, 350, 210, 35, 28, 336, 1050, 1120, 420, 48, 36, 588, 2646, 4410, 2940, 756, 63, 45, 960, 5880, 14112, 14700, 6720, 1260, 80, 55, 1485, 11880, 38808, 58212, 41580, 13860, 1980, 99, 66, 2200, 22275, 95040, 194040, 199584, 103950, 26400, 2970, 120

Offset

0,1

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

From G. C. Greubel, Feb 07 2025: (Start)

T(n, k) = binomial(n+1, n-k+1)*binomial(n+3, n-k+1).

T(2*n, n) = (1/2)*A000894(n) + (5/2)*[n=0].

Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*( (1+(-1)^n)*(-1)^(n/2)*A286033((n+4)/2) + (1-(-1)^n)*((-1)^((n+1)/2)*A000108((n+1)/2) - 1) ). (End)

Example

Triangle begins as:
   3;
   6,    8;
  10,   30,    15;
  15,   80,    90,    24;
  21,  175,   350,   210,     35;
  28,  336,  1050,  1120,    420,     48;
  36,  588,  2646,  4410,   2940,    756,     63;
  45,  960,  5880, 14112,  14700,   6720,   1260,    80;
  55, 1485, 11880, 38808,  58212,  41580,  13860,  1980,   99;
  66, 2200, 22275, 95040, 194040, 199584, 103950, 26400, 2970, 120;

Maple

for n from 0 to 10 do seq(binomial(n, i-1)*binomial(n+2, n+1-i), i=1..n ) od;

Mathematica

A124051[n_, k_]:= Binomial[n+1, n-k+1]*Binomial[n+3, n-k+1];
Table[A124051[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 07 2025 *)

Prog

(Magma)
A124051:= func< n, k | Binomial(n+1, n-k+1)*Binomial(n+3, n-k+1) >;
[A124051(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2025
(SageMath)
def A124051(n, k): return binomial(n+1, n-k+1)*binomial(n+3, n-k+1)
print(flatten([[A124051(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 07 2025

Crossrefs

Cf. A000108, A000894, A062196, A286033.

Columns k: A000217(n+2) (k=0), A002417(n+1) (k=1), A001297(n) (k=2), A105946(n-2) (k=3), A105947(n-3) (k=4), A105948(n-4) (k=5), A107319(n-5) (k=6).

Diagonals: A005563(n+1) (k=n), A033487(n) (k=n-1), A027790(n) (k=n-2), A107395(n-3) (k=n-3), A107396(n-4) (k=n-4), A107397(n-5) (k=n-5), A107398(n-6) (k=n-6), A107399(n-7) (k=n-7).

Sums: A322938(n+1) (row).

Sequence in context: A290218 A310135 A352341 * A294086 A350159 A308221

Adjacent sequences: A124048 A124049 A124050 * A124052 A124053 A124054

Keyword

nonn,tabl,easy

Author

Zerinvary Lajos, Nov 03 2006

Status

approved