A174719 -id:A174719 - cp's OEIS frontend

A174718 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 2, read by rows.

1, 1, 1, 1, -2, 1, 1, -13, -13, 1, 1, -44, -74, -44, 1, 1, -123, -278, -278, -123, 1, 1, -314, -881, -1196, -881, -314, 1, 1, -761, -2539, -4317, -4317, -2539, -761, 1, 1, -1784, -6884, -14024, -17594, -14024, -6884, -1784, 1, 1, -4087, -17884, -42412, -63874, -63874, -42412, -17884, -4087, 1

Offset: 0

Roger L. Bagula, Mar 28 2010

The row sums of this class of sequences, for varying q, is given by Sum_{k=0..n} T(n, k, q) = q^n * (n+1) + 2^n * (1 - q^n). - G. C. Greubel, Feb 09 2021

			Triangle begins as:
  1;
  1,     1;
  1,    -2,      1;
  1,   -13,    -13,      1;
  1,   -44,    -74,    -44,      1;
  1,  -123,   -278,   -278,   -123,      1;
  1,  -314,   -881,  -1196,   -881,   -314,      1;
  1,  -761,  -2539,  -4317,  -4317,  -2539,   -761,      1;
  1, -1784,  -6884, -14024, -17594, -14024,  -6884,  -1784,     1;
  1, -4087, -17884, -42412, -63874, -63874, -42412, -17884, -4087, 1;

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

Cf. A000012 (q=1), this sequence (q=2), A174719 (q=3), A174720 (q=4).

Cf. A001787, A020522.

T:= func< n,k,q | 1 + (1-q^n)*(Binomial(n,k) -1) >;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021

T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1);
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten

def T(n,k,q): return 1 + (1-q^n)*(binomial(n,k) - 1)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021

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

Sum_{k=0..n} T(n, k, 2) = 2^n *(n + 2 - 2^n) = A001787(n+1) - A020522(n). - G. C. Greubel, Feb 09 2021

Edited by G. C. Greubel, Feb 09 2021

A174720 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -14, 1, 1, -125, -125, 1, 1, -764, -1274, -764, 1, 1, -4091, -9206, -9206, -4091, 1, 1, -20474, -57329, -77804, -57329, -20474, 1, 1, -98297, -327659, -557021, -557021, -327659, -98297, 1, 1, -458744, -1769444, -3604424, -4521914, -3604424, -1769444, -458744, 1

Offset: 0

Roger L. Bagula, Mar 28 2010

The row sums of this class of sequences, for varying q, is given by Sum_{k=0..n} T(n, k, q) = q^n * (n+1) + 2^n * (1 - q^n). - G. C. Greubel, Feb 09 2021

			Triangle begins as:
  1;
  1,       1;
  1,     -14,        1;
  1,    -125,     -125,        1;
  1,    -764,    -1274,     -764,        1;
  1,   -4091,    -9206,    -9206,    -4091,        1;
  1,  -20474,   -57329,   -77804,   -57329,   -20474,        1;
  1,  -98297,  -327659,  -557021,  -557021,  -327659,   -98297,       1;
  1, -458744, -1769444, -3604424, -4521914, -3604424, -1769444, -458744, 1;

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

Cf. A000012 (q=1), A174718 (q=2), A174719 (q=3), this sequence (q=4).

Cf. A002697, A248217.

T:= func< n,k,q | 1 + (1-q^n)*(Binomial(n,k) -1) >;
[T(n,k,4): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021

T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1);
Table[T[n,k,4], {n,0,12}, {k,0,n}]//Flatten

def T(n,k,q): return 1 + (1-q^n)*(binomial(n,k) - 1)
flatten([[T(n,k,4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021

T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q=4.

Sum_{k=0..n} T(n, k, 4) = 4^n*(n+1) + 2^n*(1 - 4^n) = A002697(n+1) - A248217(n). - G. C. Greubel, Feb 09 2021

Edited by G. C. Greubel, Feb 09 2021

cp's OEIS Frontend

A174718 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 2, read by rows.

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