A218020 -id:A218020 - cp's OEIS frontend

A144018 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where sequence a_k of column k has a_k(0)=0, followed by (k+1)-fold 1 and a_k(n) shifts k places left under Euler transform.

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 9, 3, 2, 1, 1, 20, 6, 3, 2, 1, 1, 48, 10, 5, 3, 2, 1, 1, 115, 20, 8, 5, 3, 2, 1, 1, 286, 36, 14, 7, 5, 3, 2, 1, 1, 719, 72, 23, 12, 7, 5, 3, 2, 1, 1, 1842, 137, 40, 18, 11, 7, 5, 3, 2, 1, 1, 4766, 275, 69, 30, 16, 11, 7, 5, 3, 2, 1, 1, 12486, 541, 121, 47, 25, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Alois P. Heinz, Sep 07 2008

			T(5,1) = ([1,2,4]*[1,1,4] + [1]*[1]*4 + [1,2]*[1,1]*2 + [1,3]*[1,2]*1)/4 = 36/4 = 9.
Triangle begins:
    1;
    1,  1;
    2,  1,  1;
    4,  2,  1,  1;
    9,  3,  2,  1, 1;
   20,  6,  3,  2, 1, 1;
   48, 10,  5,  3, 2, 1, 1;
  115, 20,  8,  5, 3, 2, 1, 1;
  286, 36, 14,  7, 5, 3, 2, 1, 1;
  719, 72, 23, 12, 7, 5, 3, 2, 1, 1;

Alois P. Heinz, Rows n = 1..141, 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]
N. J. A. Sloane, Transforms

Columns k=1..10 give A000081, A007562, A218020, A218021, A218022, A218023, A218024, A218025, A218026, A218027.
T(2n,n) gives A000041(n).
Cf. A316074.

etrk:= proc(p) proc(n, k) option remember; `if`(n=0, 1,
         add(add(d*p(d, k), d=numtheory[divisors](j))*
         procname(n-j, k), j=1..n)/n)
       end end:
B:= etrk(T):
T:= (n, k)-> `if`(n<=k, `if`(n=0, 0, 1), B(n-k, k)):
seq(seq(T(n, k), k=1..n), n=1..14);

etrk[p_] := Module[{f}, f[n_, k_] := f[n, k] = If[n == 0, 1, (Sum[Sum[d*p[d, k], {d, Divisors[j]}]*f[n-j, k], {j, 1, n-1}] + Sum[d*p[d, k], {d, Divisors[n]}])/n]; f]; b = etrk[t]; t[n_, k_] := If[n <= k, If[n == 0, 0, 1], b[n-k, k]]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)

A316075 Sequence shifts left three places under Weigh transform with a(n) = signum(n) for n<3.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 5, 7, 10, 17, 26, 40, 65, 104, 166, 272, 442, 720, 1186, 1954, 3222, 5346, 8881, 14778, 24668, 41254, 69088, 115959, 194925, 328123, 553200, 933944, 1578614, 2671656, 4526654, 7677736, 13035809, 22154806, 37687152, 64165838

Offset: 0

Alois P. Heinz, Jun 23 2018

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Column k=3 of A316074.

Cf. A057427, A218020.

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

G.f.: x + x^2 + x^3 * Product_{n>=1} (1 + x^n)^a(n). - Ilya Gutkovskiy, May 09 2019

A346020 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, 1, 2, 3, 4, 7, 11, 17, 28, 46, 74, 124, 206, 343, 577, 976, 1649, 2808, 4792, 8200, 14073, 24228, 41782, 72246, 125164, 217262, 377784, 658072, 1148006, 2005743, 3509125, 6147422, 10782375, 18934209, 33285291, 58575080, 103181405, 181928014, 321059155

Offset: 1

Ilya Gutkovskiy, Jul 01 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..3800

Cf. A007562, A218020, A346030.

a:= proc(n) option remember; `if`(n<4, [1, 0, 1][n], add(a(n-k)*
       add(d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Jul 01 2021

nmax = 42; A[] = 0; Do[A[x] = x + x^3 Exp[Sum[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[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 42}]

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} d * a(d) ) * a(n-k).
a(n) ~ c * d^n / n^(3/2), where d = 1.82975393308934955558864748939303527364978309460948926333116466766295641... and c = 0.8335864368398390652263577663136791087027831725508605623969711758177... - Vaclav Kotesovec, Jul 06 2021

A346030 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, 1, 2, 3, 3, 6, 8, 11, 18, 26, 37, 60, 87, 132, 206, 310, 475, 742, 1130, 1759, 2737, 4236, 6618, 10348, 16139, 25350, 39767, 62456, 98401, 155047, 244570, 386639, 611298, 967874, 1534297, 2433584, 3864154, 6141560, 9766908, 15547187, 24766037, 39476846

Offset: 1

Ilya Gutkovskiy, Jul 01 2021

nonn

Cf. A007562, A218020, A346020.

a:= proc(n) option remember; `if`(n<4, signum(n-1), add(a(n-k)*
       add(d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
    end:
seq(a(n), n=1..47);  # Alois P. Heinz, Jul 01 2021

nmax = 47; A[] = 0; Do[A[x] = x^2 + x^3 Exp[Sum[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[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 47}]

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} d * a(d) ) * a(n-k).
a(n) ~ c * d^n / n^(3/2), where d = 1.646504994482771446591056040381099740295861136174688956979834656... and c = 0.8402317368556115946120005582458627329843217960728964299829... - Vaclav Kotesovec, Jul 06 2021

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A144018 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where sequence a_k of column k has a_k(0)=0, followed by (k+1)-fold 1 and a_k(n) shifts k places left under Euler transform.

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A316075 Sequence shifts left three places under Weigh transform with a(n) = signum(n) for n<3.

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A346020 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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A346030 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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