A156864 Triangle read by rows: T(n, k) = 2^k - binomial(n+1, k+1) - ((2*k-n)/(k+1)) * binomial(n+1, k).

0, -1, 1, -2, -2, 4, -3, -6, -2, 11, -4, -11, -12, 1, 26, -5, -17, -27, -19, 11, 57, -6, -24, -48, -54, -24, 36, 120, -7, -32, -76, -110, -94, -20, 92, 247, -8, -41, -112, -194, -220, -146, 8, 211, 502, -9, -51, -157, -314, -430, -398, -202, 91, 457, 1013

Offset

1,4

Comments

Row sums are zero.

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Formula

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

From G. C. Greubel, Dec 01 2019: (Start)

T(n, n) = 2^n - n - 1 = A000295(n).

Sum_{k=1..n-1} T(n,k) = - A000295(n). (End)

Example

Triangle begins as:
    0;
   -1,   1;
   -2,  -2,    4;
   -3,  -6,   -2,   11;
   -4, -11,  -12,    1,   26;
   -5, -17,  -27,  -19,   11,   57;
   -6, -24,  -48,  -54,  -24,   36,  120;
   -7, -32,  -76, -110,  -94,  -20,   92,  247;
   -8, -41, -112, -194, -220, -146,    8,  211, 502;
   -9, -51, -157, -314, -430, -398, -202,   91, 457, 1013;
  -10, -62, -212, -479, -760, -860, -664, -239, 292,  958, 2036;

Maple

b:=binomial; seq(seq( 2^k -b(n+1, k+1) -((2*k-n)/(k+1))*b(n+1, k), k=1..n), n=1..12); # G. C. Greubel, Dec 01 2019

Mathematica

T[n_, k_]:= 2^k - Binomial[n+1, k+1] - ((2*k-n)/(k+1))*Binomial[n+1, k]; Table[T[n, k], {n, 12}, {k, n}]//Flatten

Prog

(PARI) T(n, k) = 2^k -binomial(n+1, k+1) -((2*k-n)/(k+1))*binomial(n+1, k); \\ G. C. Greubel, Dec 01 2019
(Magma) B:=Binomial; [2^k -B(n+1, k+1) -((2*k-n)/(k+1))*B(n+1, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Dec 01 2019
(Sage) b=binomial; [[2^k -b(n+1, k+1) -((2*k-n)/(k+1))*b(n+1, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Dec 01 2019
(GAP) B:=Binomial;; Flat(List([1..12], n-> List([1..n], k-> 2^k - B(n+1, k+1) - ((2*k-n)/(k+1))*B(n+1, k) ))); # G. C. Greubel, Dec 01 2019

Crossrefs

Sequence in context: A328835 A076435 A257010 * A059975 A087656 A122811

Adjacent sequences: A156861 A156862 A156863 * A156865 A156866 A156867

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 17 2009

Extensions

Edited by G. C. Greubel, Dec 01 2019

Status

approved