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

1,4

Comments

Let U(n,m) = A329369(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..wt(n) + 2} k!*k^(m+1)*R(n, k)*(-1)^(wt(n) - k + 2) for n > 0, m >= 0 where wt(n) = A000120(n) and where R(n, k) are unknown coefficients.

Then T(n,k) = R(A059893(n), k).

Row n length is A000120(n) + 2.

Links

Table of n, a(n) for n=1..76.

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.

T(n, A000120(n) + 2) = A341392(n) for n >= 0.

Sum_{k=1..wt(n) + 2} k!*T(n, k)*(-1)^(wt(n) - k + 2) = A329369(n) for n >= 0 where wt(n) = A000120(n).

Sum_{k=1..wt(f(n)) + 2} k!*k^(A290255(A054429(n)) + 1)*T(A059893(f(n)), k)*(-1)^(wt(f(n)) - k + 2) = A329369(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;
  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;

Prog

(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)))

Crossrefs

Cf. A000120, A000523, A008277, A035327, A053645, A054429, A059893, A290255, A329369, A341392, A357990.

Similar tables: A358631.

Sequence in context: A076334 A348641 A014475 * A320436 A135228 A021325

Adjacent sequences: A358609 A358610 A358611 * A358613 A358614 A358615

Keyword

nonn,base,tabf

Author

Mikhail Kurkov, Nov 23 2022 [verification needed]

Status

approved