A329369 -id:A329369 - cp's OEIS frontend

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

Mikhail Kurkov, Nov 23 2022

Row n length is A000120(n) + 2.

			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;

Cf. A000120, A007814, A008277, A025480, A329369, A341392, A357990, A373183.

Similar tables: A358631.

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)))

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

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

A373183 Irregular table T(n, k), n >= 0, k > 0, read by rows with row polynomials R(n, x) such that R(2n+1, x) = xR(n, x) for n >= 0, R(2n, x) = x(R(n, x+1) - R(n, x)) for n > 0 with R(0, x) = x.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 0, 1, 3, 4, 0, 1, 2, 1, 3, 3, 0, 0, 0, 1, 7, 8, 0, 3, 4, 3, 8, 6, 0, 0, 1, 2, 7, 15, 9, 0, 1, 3, 3, 1, 4, 6, 4, 0, 0, 0, 0, 1, 15, 16, 0, 7, 8, 7, 18, 12, 0, 0, 3, 4, 17, 34, 18, 0, 3, 8, 6, 3, 11, 15, 8, 0, 0, 0, 1, 2, 31, 57, 27, 0, 7, 15

Offset: 0

Mikhail Kurkov, May 27 2024

Row n length is A000120(n) + 1.

			Irregular table begins:
  1;
  0,  1;
  1,  2;
  0,  0, 1;
  3,  4;
  0,  1, 2;
  1,  3, 3;
  0,  0, 0, 1;
  7,  8;
  0,  3, 4;
  3,  8, 6
  0,  0, 1, 2
  7, 15, 9;
  0,  1, 3, 3;
  1,  4, 6, 4;
  0,  0, 0, 0, 1;

Cf. A000120, A001511, A007814, A053645, A062253, A062254, A062255, A062383, A087734, A090996, A173018, A279209, A290255, A329369, A341392, A357990, A358612.

row(n) = my(x = 'x, A = x); forstep(i=if(n == 0, -1, logint(n, 2)), 0, -1, A = if(bittest(n, i), x*A, x*(subst(A, x, x+1) - A))); Vecrev(A/x)

Conjectured formulas: (Start)
R(2n, x) = R(n, x) + R(n - 2^f(n), x) + R(2n - 2^f(n), x) where f(n) = A007814(n) (see A329369).
b(2^m*n + q) = Sum_{i=A001511(n+1)..A000120(n)+1} T(n, i)*b(2^m*(2^(i-1)-1) + q) for n >= 0, m >= 0, q >= 0 where b(n) = A329369(n). Note that this formula is recursive for n != 2^k - 1.
R(n, x) = c(n, x)
where c(2^k - 1, x) = x^(k+1) for k >= 0,
c(n, x) = Sum_{i=0..s(n)} p(n, s(n)-i)*Sum_{j=0..i} (s(n)-j+1)^A279209(n)*binomial(i, j)*(-1)^j,
p(n, k) = Sum_{i=0..k} c(t(n) + (2^i - 1)*A062383(t(n)), x)*L(s(n), k, i) for 0 <= k < s(n) with p(n, s(n)) = c(t(n) + (2^s(n) - 1)*A062383(t(n)), x),
s(n) = A090996(n), t(n) = A087734(n),
L(n, k, m) are some integer coefficients defined for n > 0, 0 <= k < n, 0 <= m <= k that can be represented as W(n-m, k-m, m+1)
and where W(n, k, m) = (k+m)*W(n-1, k, m) + (n-k)*W(n-1, k-1, m) + [m > 1]*W(n, k, m-1) for 0 <= k < n, m > 0 with W(0, 0, m) = 1, W(n, k, m) = 0 for n < 0 or k < 0.
In particular, W(n, k, 1) = A173018(n, k), W(n, k, 2) = A062253(n, k), W(n, k, 3) = A062254(n, k) and W(n, k, 4) = A062255(n, k).
Here s(n), t(n) and A279209(n) are unique integer sequences such that n can be represented as t(n) + (2^s(n) - 1)*A062383(t(n))*2^A279209(n) where t(n) is minimal. (End)
Conjectures from Mikhail Kurkov, Jun 19 2024: (Start)
T(n, k) = d(n, 1, A000120(n) - k + 2) where d(n, m, k) = (m+1)^g(n)*d(h(n), m+1, k) - m^(g(n)+1)*d(h(n), m, k-1) for n > 0, m > 0, k > 0 with d(n, m, 0) = 0 for n >= 0, m > 0, d(0, m, k) = [k <= m]*abs(Stirling1(m, m-k+1)) for m > 0, k > 0, g(n) = A290255(n) and where h(n) = A053645(n). In particular, d(n, 1, 1) = A341392(n).
Sum_{i=A001511(n+1)..wt(n)+k} d(n, k, wt(n)-i+k+1)*A329369(2^m*(2^(i-1)-1) + q) = k!*A357990(2^m*n + q, k) for n >= 0, k > 0, m >= 0, q >= 0 where wt(n) = A000120(n).
If we change R(0, x) to Product_{i=0..m-1} (x+i), then for resulting irregular table U(n, k, m) we have U(n, k, m) = d(n, m, A000120(n) - k + m + 1).
T(n, k) = (-1)^(wt(n)-k+1)*Sum_{i=1..wt(n)-k+3} Stirling1(wt(n)-i+3, k+1)*A358612(n, wt(n)-i+3) for n >= 0, k > 0 where wt(n) = A000120(n). (End)
Conjecture: T(2^m*(2k+1), q) = (-1)^(wt(k)-q)*Sum_{i=q..wt(k)+2} Stirling1(i,q)*A358612(k,i)*i^m for m >= 0, k >= 0, q > 0 where wt(n) = A000120(n). - Mikhail Kurkov, Jan 17 2025

A344947 Number of open tours by a biased rook on a specific A070941(n) X 1 board, which ends on a black cell, where cells are colored white or black according to the binary representation of 2n.

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 8, 18, 1, 4, 8, 18, 16, 36, 54, 96, 1, 4, 8, 18, 16, 36, 54, 96, 32, 72, 108, 192, 162, 288, 384, 600, 1, 4, 8, 18, 16, 36, 54, 96, 32, 72, 108, 192, 162, 288, 384, 600, 64, 144, 216, 384, 324, 576, 768, 1200, 486, 864, 1152, 1800, 1536, 2400

Offset: 0

Mikhail Kurkov, Jun 03 2021 [verification needed]

nonn

A cell is colored white if the binary digit is 0 and a cell is colored black if the binary digit is 1. A biased rook on a white cell moves only to the left and otherwise moves in any direction.

Cf. A053645, A070941, A284005, A329369, A329718.

a(n) = A284005(2*A053645(n)) for n > 0 with a(0) = 1.

A379817 Irregular table T(n, k), n >= 0, k >= 0, read by rows such that T(n,k) = f(n,k)/f(2^k-1,k) where f(n,k) is defined in Comments.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 3, 1, 7, 4, 3, 7, 2, 7, 12, 3, 1, 7, 6, 1, 15, 8, 7, 15, 4, 17, 26, 6, 3, 17, 13, 2, 31, 42, 9, 7, 31, 21, 3, 15, 50, 30, 4, 1, 15, 25, 10, 1, 31, 16, 15, 31, 8, 37, 54, 12, 7, 37, 27, 4, 69, 88, 18, 17, 69, 44, 6, 37, 112, 63, 8, 3, 37, 56

Offset: 0

Mikhail Kurkov, Jan 03 2025

Here f(n,k) = b(2^k*(2n+1)) - Sum_{j=1..k} b(2^(j-1)*(2n+1))*R(k,j) for n >= 0, k >= 0 where b(n) = A329369(n) and where R(k,j) is the unique solution to b(2^k*(2^i-1)) = Sum_{j=1..k} b(2^(j-1)*(2^i-1))*R(k,j) for k > 0, 1 <= i <= k.

Row n length is A000120(n) + 1.

			Irregular table begins:
   1;
   1,  1;
   3,  2;
   1,  3,  1;
   7,  4;
   3,  7,  2;
   7, 12,  3;
   1,  7,  6,  1;
  15,  8;
   7, 15,  4;
  17, 26,  6;
   3, 17, 13,  2;
  31, 42,  9;
   7, 31, 21,  3;
  15, 50, 30,  4;
   1, 15, 25, 10, 1;

Cf. A000120, A341392, A329369.

upto(n) = my(A, v1); v1 = vector(n+1, i, 0); v1[1] = 1; for(i=1, n, v1[i+1] = v1[i\2+1] + if(i%2, 0, A = 1 << valuation(i/2, 2); v1[i/2-A+1] + v1[i-A+1])); v1 \\ from A329369
R(k) = my(v1, M1, M2); v1 = upto(2^k*(2^k-1)); M1 = matrix(k, k, i, j, v1[2^(j-1)*(2^i-1)+1]); M2 = matrix(k, 1, i, j, v1[2^k*(2^i-1)+1]); M1 = matsolve(M1, M2)
row(n) = my(A = hammingweight(n), v1, v2, v3); v1 = upto(2^A*(2*n+1)); v2 = vector(A, i, R(i)); v3 = vector(A, i, (v1[2^i*(2*n+1)+1] - sum(j=1, i, v1[2^(j-1)*(2*n+1)+1]*v2[i][j,1]))/(v1[2^i*(2*(2^i-1)+1)+1] - sum(j=1, i, v1[2^(j-1)*(2*(2^i-1)+1)+1]*v2[i][j,1]))); concat(v1[n+1], v3)

Conjectures: (Start)
f(2^k-1,k) = ((k+1)!)^2 for k >= 0.
R(k,j) = -Stirling1(k+2, j+1) for k > 0, 1 <= j <= k.
T(2^n-1, k) = Stirling2(n+1, k+1) for n >= 0, 0 <= k <= n.
T(n,k) = c(n,wt(n)-k) for n >= 0, 0 <= k <= wt(n) where c(2n+1,k) = c(n,k) + (wt(n)-k+2)*c(n,k-1), c(2n,k) = (wt(n)-k+1)*c(2n+1,k) for n > 0, k > 0 with c(n,0) = A341392(n) for n >= 0, c(0,k) = 0 for k > 0 and where wt(n) = A000120(n). (End)

A379818 a(2n+1) = a(n) for n >= 0, a(2n) = a(n) + a(n - 2^f(n)) + a(2n - 2^f(n)) + a(A025480(n-1)) for n > 0 with a(0) = 1 where f(n) = A007814(n).

Original entry on oeis.org

1, 1, 4, 1, 10, 4, 10, 1, 22, 10, 28, 4, 49, 10, 22, 1, 46, 22, 64, 10, 118, 28, 64, 4, 190, 49, 118, 10, 190, 22, 46, 1, 94, 46, 136, 22, 256, 64, 148, 10, 424, 118, 292, 28, 478, 64, 136, 4, 661, 190, 478, 49, 796, 118, 256, 10, 1177, 190, 424, 22, 661, 46

Offset: 0

Mikhail Kurkov, Jan 03 2025

Cf. A007814, A025480, A329369, A341392, A347205.

upto(n) = my(A, v1); v1 = vector(n+1, i, 0); v1[1] = 1; for(i=1, n, v1[i+1] = v1[i\2+1] + if(i%2, 0, A = 1 << valuation(i/2, 2); v1[i/2-A+1] + v1[i-A+1] + v1[i\(4*A)+1])); v1

Conjecture: a(2^m*(2k+1)) = Sum_{j=0..m} (binomial(m+2, j+1) - binomial(m, j))*a(2^j*k) for m >= 0, k >= 0 with a(0) = 1.

A363417 a(n) = Sum_{j=0..2^n - 1} b(j) for n >= 0 where b(n) = (A023416(n) + 1)b(A053645(n)) + [A036987(n) = 0]b(A266341(n)) for n > 0 with b(0) = 1.

Original entry on oeis.org

1, 2, 6, 23, 106, 566, 3415, 22872, 167796, 1334596, 11414192, 104270906, 1011793389, 10379989930, 112134625986, 1271209859403, 15077083642150, 186588381229340, 2403775013224000, 32168379148440968, 446341838086450308, 6410107231501731012, 95136428354649665256

Offset: 0

Mikhail Kurkov, Jun 11 2023 [verification needed]

nonn

Note that [A036987(n) = 0]*b(A266341(n)) is the same as max((1 - T(n, j))*b(A053645(n) + 2^j*(1 - T(n, j))) | 0 <= j <= A000523(n)) where T(n, k) = floor(n/2^k) mod 2.

In fact b(n) is a generalization of A347205 just as A329369 is a generalization of A341392.

Cf. A000523, A023416, A036987, A053645, A266341.

Similar recurrences: A284005, A329369, A341392, A347205.

A063250(n)=my(L=logint(n, 2), A=0); for(i=0, L, my(B=n\2^(L-i)+1); A++; A-=logint(B, 2)==valuation(B, 2)); A
upto(n)=my(v, v1); v=vector(2^n, i, 0); v[1]=1; v1=vector(n+1, i, 0); v1[1]=1; for(i=1, #v-1, my(L=logint(i, 2), A=i - 2^L, B=A063250(i)); v[i+1]=(L - hammingweight(i) + 2)*v[A+1] + if(B>0, v[A + 2^(B-1) + 1])); for(i=1, n, v1[i+1]=v1[i] + sum(j=2^(i-1)+1, 2^i, v[j])); v1

cp's OEIS Frontend

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.

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A373183 Irregular table T(n, k), n >= 0, k > 0, read by rows with row polynomials R(n, x) such that R(2n+1, x) = x*R(n, x) for n >= 0, R(2n, x) = x*(R(n, x+1) - R(n, x)) for n > 0 with R(0, x) = x.

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A344947 Number of open tours by a biased rook on a specific A070941(n) X 1 board, which ends on a black cell, where cells are colored white or black according to the binary representation of 2n.

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A379817 Irregular table T(n, k), n >= 0, k >= 0, read by rows such that T(n,k) = f(n,k)/f(2^k-1,k) where f(n,k) is defined in Comments.

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A379818 a(2n+1) = a(n) for n >= 0, a(2n) = a(n) + a(n - 2^f(n)) + a(2n - 2^f(n)) + a(A025480(n-1)) for n > 0 with a(0) = 1 where f(n) = A007814(n).

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A363417 a(n) = Sum_{j=0..2^n - 1} b(j) for n >= 0 where b(n) = (A023416(n) + 1)*b(A053645(n)) + [A036987(n) = 0]*b(A266341(n)) for n > 0 with b(0) = 1.

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A373183 Irregular table T(n, k), n >= 0, k > 0, read by rows with row polynomials R(n, x) such that R(2n+1, x) = xR(n, x) for n >= 0, R(2n, x) = x(R(n, x+1) - R(n, x)) for n > 0 with R(0, x) = x.

A363417 a(n) = Sum_{j=0..2^n - 1} b(j) for n >= 0 where b(n) = (A023416(n) + 1)b(A053645(n)) + [A036987(n) = 0]b(A266341(n)) for n > 0 with b(0) = 1.