A358631 Irregular table T(n, k), n >= 0, k > 0, read by rows of extended (due to binary expansion of n) Stirling numbers of the first kind.

1, 1, 2, 3, 1, 4, 5, 1, 6, 11, 6, 1, 6, 7, 1, 12, 20, 9, 1, 18, 26, 9, 1, 24, 50, 35, 10, 1, 8, 9, 1, 18, 29, 12, 1, 30, 41, 12, 1, 48, 94, 59, 14, 1, 36, 47, 12, 1, 72, 130, 71, 14, 1, 96, 154, 71, 14, 1, 120, 274, 225, 85, 15, 1, 10, 11, 1, 24, 38, 15, 1, 42

Offset: 0

Author

Mikhail Kurkov, Nov 24 2022

Keywords

• nonn
• base
• tabf

Comments

Row n length is A000120(n) + 2.

Examples

			Irregular table begins:
    1,   1;
    2,   3,   1;
    4,   5,   1;
    6,  11,   6,  1;
    6,   7,   1;
   12,  20,   9,  1;
   18,  26,   9,  1;
   24,  50,  35, 10,  1;
    8,   9,   1;
   18,  29,  12,  1;
   30,  41,  12,  1;
   48,  94,  59, 14,  1;
   36,  47,  12,  1;
   72, 130,  71, 14,  1;
   96, 154,  71, 14,  1;
  120, 274, 225, 85, 15, 1;
		

Crossrefs

Cf. A000120, A053645, A052852, A063250, A132393, A290255, A347205.

Similar tables: A358612.

Programs

PARI

b1(n)=if(n>0, my(A=n - 2^logint(n, 2)); if(A>0, logint(A, 2) + 1))
b2(n)=if(n>0, my(A=b1(3*2^logint(n, 2) - n - 1)); n + if(A>0, 2^(A-1)))
P(n,k)=if(n==0 || k==1, (n==0 && k<3) + (k==1 && n>0), my(L=logint(n, 2), A=n - 2^L); (hammingweight(A) + 2)*P(A, k-1)*(L - b1(n) + 1) + P(b2(A), k))
T(n, k)=my(A=hammingweight(n)); if(k<=(A + 2), P(n, A - k + 3))

Formula

T(n, k) = P(n, wt(n) - k + 3) for n >= 0, 0 < k <= wt(n) + 2 where wt(n) = A000120(n).

P(n, 1) = 1 for n > 0 with P(0, 1) = P(0, 2) = 1.

P(n, k) = (A000120(q(n)) + 2)*P(q(n), k-1)*(A290255(n) + 1) + P(s(q(n)), k) for n > 0, k > 1 where q(n) = A053645(n) and where s(n) = n + [A063250(n) > 0]*2^(A063250(n) - 1).

T(2^n - 1, k) = abs(Stirling1(n+2, k)) for n >= 0, k > 0.

Conjectures: (Start)

T(n, 1) = (A000120(n) + 1)!*A347205(n) for n >= 0.

Sum_{k=1..A000120(n) + 2} T(n, k)*(-1)^k = 0 for n >= 0.

Sum_{k=0..2^n - 1} Sum_{j=1..A000120(k) + 2} T(k, j) = 2*A052852(n+1) for n >= 0.

Sum_{i=1..wt(k) + 2} m^(i-1)*T(k, i) = (wt(k) + 1)!*A347205(2^m*(2k+1)) for m >= 0, k >= 0 where wt(n) = A000120(n). (End)

Extensions

Offset corrected by Mikhail Kurkov, Nov 07 2024