A176152 Triangle, read by rows, T(n, k) = 2*binomial(n, k)*binomial(n+1, k)/(k+1) - (k! - n! + (n-k)!).

1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 37, 60, 37, 1, 1, 125, 212, 212, 125, 1, 1, 641, 904, 1058, 904, 641, 1, 1, 4375, 5310, 5990, 5990, 5310, 4375, 1, 1, 35351, 40270, 42546, 43800, 42546, 40270, 35351, 1, 1, 322649, 358918, 367194, 373320, 373320, 367194, 358918, 322649, 1

Offset

0,5

Comments

Row sums are: {1, 2, 8, 32, 136, 676, 4150, 31352, 280136, 2844164, 31958544, ...}.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n, k) = 2*binomial(n, k)*binomial(n+1, k)/(k+1) - (k! - n! + (n-k)!).

Example

Triangle begins as:
  1;
  1,      1;
  1,      6,      1;
  1,     15,     15,      1;
  1,     37,     60,     37,      1;
  1,    125,    212,    212,    125,      1;
  1,    641,    904,   1058,    904,    641,      1;
  1,   4375,   5310,   5990,   5990,   5310,   4375,      1;
  1,  35351,  40270,  42546,  43800,  42546,  40270,  35351,      1;
  1, 322649, 358918, 367194, 373320, 373320, 367194, 358918, 322649, 1;

Maple

T:= 2*binomial(n, k)*binomial(n+1, k)/(k+1) -(k! -n! +(n-k)!); seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 23 2019

Mathematica

T[n_, k_] = 2*Binomial[n, k]*Binomial[n+1, k]/(k+1) -(k! -n! +(n-k)!); Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Nov 23 2019 *)

Prog

(PARI) T(n, k) = 2*binomial(n, k)*binomial(n+1, k)/(k+1) -(k!-n!+(n-k)!); \\ G. C. Greubel, Nov 23 2019
(Magma) F:=Factorial; [2*Binomial(n, k)*Binomial(n+1, k)/(k+1) - (F(k) - F(n) + F(n-k)): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 23 2019
(Sage) f=factorial; b=binomial; [[2*b(n, k)*b(n+1, k)/(k+1) -f(k) +f(n) - f(n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 23 2019
(GAP) F:=Factorial;; B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> 2*B(n, k)*B(n+1, k)/(k+1) -F(k) +F(n) -F(n-k) ))); # G. C. Greubel, Nov 23 2019

Crossrefs

Cf. A155170.

Sequence in context: A154980 A166344 A146766 * A146958 A154653 A109001

Adjacent sequences: A176149 A176150 A176151 * A176153 A176154 A176155

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 10 2010

Extensions

Edited by G. C. Greubel, Nov 23 2019

Status

approved