A308737 Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is the 1-tiered binomial coefficient.

1, 1, 3, 1, 8, 7, 1, 17, 31, 15, 1, 34, 96, 94, 31, 1, 67, 258, 382, 253, 63, 1, 132, 645, 1280, 1275, 636, 127, 1, 261, 1545, 3845, 5115, 3831, 1531, 255, 1, 518, 3598, 10766, 17920, 17906, 10738, 3578, 511, 1, 1031, 8212, 28700, 57358, 71666, 57316, 28652, 8185, 1023

Offset: 0

Author

Michel Marcus, Jun 21 2019

Keywords

• nonn
• tabl

Examples

			From _Petros Hadjicostas_, Jul 07 2020: (Start)
Square array for (N,M)_1 of 1-tiered binomial coefficients (N, M >= 0):
  1/2,   3/4,    7/8,     15/16,   31/32,     63/64,   127/128, ...
  1/4,    1,    31/16,    47/16,  253/64,    159/32,  1531/256, ...
  1/8,  17/16,    3,     191/32, 1275/128,  3831/256,  5369/256, ...
  1/16, 17/16, 129/32,     10,   5115/256,  8953/256, 14329/256, ...
  1/32, 67/64, 645/128, 3845/256,   35,    35833/512, 129003/1024, ...
  ... (End)
Triangle (n-k,k)_1 of 1-tiered binomial coefficients (n >= 0 and k = 0..n):
  1/2,
  1/4,    3/4,
  1/8,     1,    7/8,
  1/16,  17/16, 31/16, 15/16,
  1/32,  17/16,   3,   47/16, 31/32,
  ...
Scaled triangle T(n,k) after multiplying each row by 2^(n+1):
  1,
  1,  3,
  1,  8,  7,
  1, 17, 31, 15,
  1, 34, 96, 94, 31,
  ...
		

Links

• Michael E. Hoffman, (Poly)logarithmic Integrals and Multiple Zeta Values, Number Theory Talk, Max-Planck-Institut für Mathematik, Bonn, 20 June 2018. See Slide 29.
• Michael E. Hoffman and Markus Kuba, Logarithmic integrals, zeta values, and tiered binomial coefficients, arXiv:1906.08347 [math.CO], 2019-2020. See Example 4 on pp. 12-13.

Crossrefs

Cf. A007318 (Pascal triangle: 0-tiered binomial coefficient), A038208, A145071 (column k = 1).

Programs

Mathematica

rows = 10;
cc = CoefficientList[# + O[y]^rows, y]& /@ CoefficientList[(1-x)/((1-x-y)* (2-x-y)) + O[x]^rows, x];
T[n_, m_, 1] := cc[[n-m+1, m+1]];
Table[2^(n+1) Table[T[n, m, 1], {m, 0, n}], {n, 0, rows-1}] (* Jean-François Alcover, Jun 21 2019 *)

PARI

T(n,m) = if ((n==0) && (m==0), 1/2, binomial(n+m-1, m-1) - (binomial(n+m,n)/2 - binomial(n+m-1,n-1))/2^(n+m));
TT(n, k) = T(n-k, k);
tabls(nn) = for (n=0, nn, for (k=0, n, print1(2^(n+1)*TT(n, k), ", ")));

Formula

(N,M)1 = ((N-1,M)_1 + (N,M-1)_1 + (N,M-1)_0)/2 for N, M >= 0 and N + M > 0 with initial conditions: (N,-1)_i = 0 = (-1,M)_i for i = 0, 1; (0,0)_1 = 1/2; and (N,M)_0 = binomial(N+M, M) for N, M >= 0. (This is the recurrence for the square array presentation of the regular triangle.) [Edited by _Petros Hadjicostas, Jul 06 2020]

(N,M)1 = binomial(N+M-1, M-1) - (binomial(N+M, N)/2 - binomial(N+M-1, N-1))/2^(N+M) for N,M >= 0 and N + M > 0 with (0,0)_1 = 1/2. - _Petros Hadjicostas, Jul 06 2020

G.f. for (N,M)1: (1-x)/((1-x-y)*(2-x-y)). - _Jean-François Alcover, Jun 21 2019

Scaled coefficients satisfy T(n,0) = 1 for n >= 0 and T(n,k) = T(n-1,k) + T(n-1,k-1) + 2^n*C(n-1,k-1) for n >= k+1 >= 1. - Charlie Neder, Jun 21 2019 [Corrected by Petros Hadjicostas, Jul 06 2020]

From Petros Hadjicostas, Jul 07 2020: (Start)

(N,M)_1 + (M,N)_1 = (N,M)_0 = binomial(N+M, N) for N, M >= 0.

(n-k,k)_1 + (k, n-k)_1 = binomial(n,k) for n >= k >= 0.

T(n,k) + T(n,n-k) = 2^(n+1)*binomial(n,k) = 2*A038208(n,k) for n >= k >= 0.

T(n,k) = 2^(n + 1)*binomial(n-1, k-1) + 2*binomial(n-1,k) - binomial(n,k) for n >= k >= 0 and (n,k) <> (0,0) with T(0,0) = 1.

G.f. for T(n,k): (1 - 2*x)/((1 - 2*x*(1 + y))*(1 - x*(1 + y))). (End)

Extensions

Name edited by Petros Hadjicostas, Jul 07 2020