A358612 Irregular table T(n, k), n >= 0, k > 0, read by rows of extended (due to binary expansion of n) Stirling numbers of the second kind.
1, 1, 1, 3, 1, 1, 5, 2, 1, 7, 6, 1, 1, 9, 4, 1, 11, 11, 2, 1, 13, 15, 3, 1, 15, 25, 10, 1, 1, 17, 8, 1, 19, 21, 4, 1, 21, 28, 6, 1, 23, 44, 19, 2, 1, 25, 39, 9, 1, 27, 58, 27, 3, 1, 29, 68, 34, 4, 1, 31, 90, 65, 15, 1, 1, 33, 16, 1, 35, 41, 8, 1, 37, 54, 12, 1
Offset: 0
Examples
Irregular table begins: 1, 1; 1, 3, 1; 1, 5, 2; 1, 7, 6, 1; 1, 9, 4; 1, 11, 11, 2; 1, 13, 15, 3; 1, 15, 25, 10, 1; 1, 17, 8; 1, 19, 21, 4; 1, 21, 28, 6; 1, 23, 44, 19, 2; 1, 25, 39, 9; 1, 27, 58, 27, 3; 1, 29, 68, 34, 4; 1, 31, 90, 65, 15, 1;
Crossrefs
Programs
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PARI
T(n, k)=if(n==0 || k==1, (n==0 && k<3) + (k==1 && n>0), k*T(n\2, k) + T(n\2, k-1) - if(n%2==0, (T(n, k-1) + T(n\2,k-1))/(k-1)))
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PARI
row(n) = my(A, v1, v2); v1 = [1, 1]; if(n == 0, v1, forstep(i=logint(n, 2), 0, -1, A = bittest(n, i); v2 = vector(#v1+A, i, 0); v2[1] = 1; for(j=2, #v2, v2[j] = j*if(j==#v1+1, 0, v1[j]) + v1[j-1] - if(A, 0, (v2[j-1] + v1[j-1])/(j-1))); v1 = v2); v1) \\ Mikhail Kurkov, Apr 30 2024
Formula
T(n, 1) = 1 for n > 0 with T(0, 1) = T(0, 2) = 1.
T(2n+1, k) = k*T(n, k) + T(n, k-1) for n >= 0, k > 1.
T(2n, k) = k*T(n, k) + T(n, k-1) - (T(2n, k-1) + T(n, k-1))/(k-1) for n > 0, k > 1.
T(2^n - 1, k) = Stirling2(n+2, k) for n >= 0, k > 0.
T(n, 2) = 2n+1 for n >= 0.
Conjectured formulas: (Start)
Sum_{i=1..wt(k) + 2} i!*i^m*T(k, i)*(-1)^(wt(k) - i + 2) = A329369(2^m*(2k+1)) for m >= 0, k >= 0 where wt(n) = A000120(n). (End)
Conjecture: T(n, k) = (k-1)^g(n)*T(h(n), k-1) + k^(g(n)+1)*T(h(n), k) for n > 0, k > 1 with T(n, 1) = T(0, 2) = 1 where g(n) = A007814(n) and where h(n) = A025480(n-1). - Mikhail Kurkov, Jun 21 2024
Comments