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

1,3

Comments

Let U(n,m) = A347205(2*(A054429(n + 2^(A000523(n) + 1)) + 2^(A000523(n) + 2)*(2^m - 1))).

Experiments with WolframAlpha lead us to conjecture (which we subsequently check on a large number of values) that

U(n,m) = Sum_{k=1..A000120(n) + 2} (m+1)^(k-1)*R(n, k) for n > 0, m >= 0 where R(n, k) are unknown coefficients.

Then T(n,k) = (A000120(n) + 1)!*R(A059893(n), k).

Row n length is A000120(n) + 2.

Links

Table of n, a(n) for n=1..72.

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.

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_{k=1..wt(f(n)) + 2} (A290255(A054429(n)) + 1)^(k-1)*T(A059893(f(n)), k) = (wt(f(n)) + 1)!*A347205(2n) for n > 0, A053645(n+1) > 0 where wt(n) = A000120(n) and where f(n) = A035327(n).

Example

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;

Prog

(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))

Crossrefs

Cf. A000120, A000523, A035327, A053645, A054429, A059893, A063250, A132393, A290255, A347205.

Similar tables: A358612.

Sequence in context: A038802 A092942 A229137 * A358106 A282510 A131225

Adjacent sequences: A358628 A358629 A358630 * A358632 A358633 A358634

Keyword

nonn,base,tabf

Author

Mikhail Kurkov, Nov 24 2022 [verification needed]

Status

approved