A316075 -id:A316075 - cp's OEIS frontend

A316074 Sequence a_k of column k shifts left k places under Weigh transform and equals signum(n) for n=1, 1<=k<=n, read by rows.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 6, 2, 2, 1, 1, 1, 12, 4, 2, 2, 1, 1, 1, 25, 6, 3, 2, 2, 1, 1, 1, 52, 10, 5, 3, 2, 2, 1, 1, 1, 113, 17, 7, 4, 3, 2, 2, 1, 1, 1, 247, 29, 10, 6, 4, 3, 2, 2, 1, 1, 1, 548, 51, 17, 8, 5, 4, 3, 2, 2, 1, 1, 1, 1226, 89, 26, 12, 7, 5, 4, 3, 2, 2, 1, 1, 1

Offset: 1

Alois P. Heinz, Jun 23 2018

			Triangle T(n,k) begins:
    1;
    1,  1;
    1,  1, 1;
    2,  1, 1, 1;
    3,  2, 1, 1, 1;
    6,  2, 2, 1, 1, 1;
   12,  4, 2, 2, 1, 1, 1;
   25,  6, 3, 2, 2, 1, 1, 1;
   52, 10, 5, 3, 2, 2, 1, 1, 1;
  113, 17, 7, 4, 3, 2, 2, 1, 1, 1;

Alois P. Heinz, Rows n = 1..200, flattened
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Columns k=1-10 give: A004111, A007560, A316075, A316076, A316077, A316078, A316079, A316080, A316081, A316082.
T(2n,n) gives A000009.
Cf. A057427, A144018, A316101.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(T(i, k), j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
T:= (n, k)-> `if`(n

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[T[i, k], j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
T[n_, k_] := If[n < k, Sign[n], b[n - k, n - k, k]];
Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)

A218020 Shifts 3 places left under Euler transform with a(0)=0 and a(n)=1 for n < 3.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 5, 8, 14, 23, 40, 69, 121, 212, 378, 672, 1208, 2177, 3946, 7173, 13104, 23995, 44103, 81261, 150149, 278054, 516141, 959952, 1788950, 3339656, 6245177, 11696510, 21938857, 41206395, 77496891, 145926374, 275098857, 519181163, 980848600

Offset: 0

Alois P. Heinz, Oct 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Column k=3 of A144018.

Cf. A316075.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1,
      (add(add(d*a(d), d=divisors(j)) *b(n-j), j=1..n))/n)
    end:
a:= n-> `if`(n<3, signum(n), b(n-3)):
seq(a(n), n=0..40);

b[n_] := b[n] = If[n == 0, 1, (Sum[Sum[d*a[d], {d, Divisors[j]}]*b[n - j], {j, 1, n }])/n]; a[0] = 0; a[1] = a[2] = 1; a[n_] := b[n - 3]; Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 1.964293016979213611214370656... and c = 0.8776048696248050091050307... . - Vaclav Kotesovec, Jun 23 2014

G.f.: x + x^2 + x^3 / Product_{n>=1} (1 - x^n)^a(n). - Ilya Gutkovskiy, May 08 2019

A346031 G.f. A(x) satisfies: A(x) = x + x^3 * exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 1, 1, 4, 4, 3, 9, 14, 12, 22, 43, 49, 66, 130, 186, 234, 406, 663, 884, 1362, 2303, 3347, 4884, 8049, 12478, 18240, 28853, 46075, 69163, 106470, 170305, 262853, 401773, 635780, 998609, 1536093, 2405345, 3801601, 5910267, 9212253, 14548179, 22818301

Offset: 1

Ilya Gutkovskiy, Jul 01 2021

nonn

Cf. A007560, A316075, A346032.

a:= proc(n) option remember; `if`(n<4, [1, 0, 1][n], add(a(n-k)*add(
     (-1)^(k/d+1)*d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
    end:
seq(a(n), n=1..48);  # Alois P. Heinz, Jul 01 2021

nmax = 48; A[] = 0; Do[A[x] = x + x^3 Exp[Sum[(-1)^(k + 1) A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[2] = 0; a[3] = 1; a[n_] := a[n] = (1/(n - 3)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 48}]

G.f.: x + x^3 * Product_{n>=1} (1 + x^n)^a(n).

a(1) = 1, a(2) = 0, a(3) = 1; a(n) = (1/(n - 3)) * Sum_{k=1..n-3} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k).

A346032 G.f. A(x) satisfies: A(x) = x^2 + x^3 * exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 0, 2, 1, 1, 4, 2, 4, 9, 4, 14, 20, 15, 43, 48, 55, 127, 127, 199, 363, 379, 684, 1048, 1229, 2263, 3100, 4163, 7288, 9558, 14231, 23222, 30673, 48404, 74113, 101631, 163048, 239282, 343196, 545318, 785139, 1169148, 1818866, 2619072, 3991888, 6079434

Offset: 1

Ilya Gutkovskiy, Jul 01 2021

nonn

Cf. A007560, A316075, A346031.

a:= proc(n) option remember; `if`(n<4, signum(n-1), add(a(n-k)*add(
     (-1)^(k/d+1)*d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 01 2021

nmax = 50; A[] = 0; Do[A[x] = x^2 + x^3 Exp[Sum[(-1)^(k + 1) A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 0; a[2] = 1; a[3] = 1; a[n_] := a[n] = (1/(n - 3)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 50}]

G.f.: x^2 + x^3 * Product_{n>=1} (1 + x^n)^a(n).

a(1) = 0, a(2) = 1, a(3) = 1; a(n) = (1/(n - 3)) * Sum_{k=1..n-3} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k).

cp's OEIS Frontend

A316074 Sequence a_k of column k shifts left k places under Weigh transform and equals signum(n) for n=1, 1<=k<=n, read by rows.

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A218020 Shifts 3 places left under Euler transform with a(0)=0 and a(n)=1 for n < 3.

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A346031 G.f. A(x) satisfies: A(x) = x + x^3 * exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).

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A346032 G.f. A(x) satisfies: A(x) = x^2 + x^3 * exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).

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