OFFSET
0,3
COMMENTS
Equivalently, T(n,k) are the unique integer coefficients such that Sum_{k=0..n} T(n,k)*A125790(k,m)*(-1)^(n+k)/2^(n*k) = m^n for all n >= 0, m >= 0. - Mikhail Kurkov, Jan 12 2026
LINKS
MathOverflow, Recurrence for columns of A125790, 2025.
MathOverflow, Proof of a recurrent formula if the closed form is known, 2025.
MathOverflow, Pair of identities for A125790, 2026.
FORMULA
T(n,k) = Sum_{j=0..2^k-1} (2*j+1)^n*(-1)^(A000120(j)+k).
T(n,k) = Sum_{j=0..n-1} binomial(n,j)*2^(k*(n-j))*T(j,k-1) for k > 0 with T(n,0) = 1.
T(n,k) = 2*Sum_{j=0..floor((n-k)/2)} binomial(n,k+2*j)*2^(k*(n-k-2*j))*T(k+2*j,k-1) for k > 0 with T(n,0) = 1.
E.g.f. for column k: exp(x)*Product_{j=1..k} (exp(2^j*x) - 1).
EXAMPLE
Triangle begins:
1;
1, 2;
1, 8, 16;
1, 26, 192, 384;
1, 80, 1696, 12288, 24576;
1, 242, 13440, 272640, 1966080, 3932160;
1, 728, 101296, 5222400, 104816640, 754974720, 1509949440;
1, 2186, 743232, 92663424, 4693032960, 93943234560, 676457349120, 1352914698240;
MATHEMATICA
T[n_, k_]:=Sum[(2j+1)^n*(-1)^(DigitCount[j, 2, 1]+k), {j, 0, 2^k-1}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}]//Flatten (* James C. McMahon, Dec 30 2025 *)
PROG
(PARI) rows(n) = my(v1 = vector(n+1, i, vector(i, j, 0)), v2 = v1); v1[1][1] = 1; v2[1][1] = 1; for(i=1, n, v1[i+1][1] = 1; v2[i+1][1] = 1; v2[i+1][i+1] = 1); for(i=2, n, for(j=1, i-1, v2[i+1][j+1] = v2[i][j+1] + v2[i][j])); for(i=1, n, for(j=i, n, v1[j+1][i+1] = 2*sum(k=0, (j-i)\2, v2[j+1][i+2*k+1]*v1[i+2*k+1][i]*2^(i*(j-i-2*k))))); v1
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Mikhail Kurkov, Dec 23 2025
STATUS
approved