OFFSET
0,3
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.
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)
EXAMPLE
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,
...
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 *)
PROG
(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), ", ")));
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Michel Marcus, Jun 21 2019
EXTENSIONS
Name edited by Petros Hadjicostas, Jul 07 2020
STATUS
approved