A051573 -id:A051573 - cp's OEIS frontend

A051603 Expansion of square of g.f. for A051573.

1, 2, 3, 6, 11, 26, 58, 142, 351, 890, 2282, 5948, 15624, 41442, 110703, 297676, 804892, 2187490, 5971392, 16366734, 45021391, 124253828, 343956858, 954760502, 2656946827, 7411140120, 20716895918, 58027609028, 162837485745

Offset: 0

David Broadhurst

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A052250, A051573.

(* r = A000081 *) r[n_] := r[n] = If[n<2, n, Sum[DivisorSum[j, #*r[#]&] * r[n-j], {j, 1, n-1}]/(n-1)]; b[n_] := b[n] = -If[n<0, 1, Sum[b[n-i] * r[i+1], {i, 1, n+1}]]; B[n_] := Sum[b[i]*x^i, {i, 0, n}]; T[n_, k_] := Coefficient[B[n]^k, x, n-k]; a[n_] := T[n+2, 2]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 23 2017, after Alois P. Heinz *)

A052250 Triangle T(n,k) (n >= 1, k >= 1) giving dimension of bigrading of Hopf algebra of rooted trees.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 3, 6, 6, 4, 1, 8, 11, 13, 10, 5, 1, 16, 26, 27, 24, 15, 6, 1, 41, 58, 63, 55, 40, 21, 7, 1, 98, 142, 148, 132, 100, 62, 28, 8, 1, 250, 351, 363, 322, 251, 168, 91, 36, 9, 1, 631, 890, 912, 804, 635, 444, 266, 128, 45, 10, 1, 1646, 2282, 2330, 2051

Offset: 1

David Broadhurst, Feb 05 2000

			Triangle begins
  1;
  1, 1;
  1, 2, 1;
  2, 3, 3, 1;
  3, 6, 6, 4, 1;

Alois P. Heinz, Rows n = 1..141, flattened
D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization...

First few columns give A051573, A051603, A052251, A052252.

Row sums give A000081(n+1). - Alois P. Heinz, Oct 23 2009

with(numtheory): A81:= proc(n) option remember; `if`(n<2, n, (add(add(d*A81(d), d=divisors(j)) *A81(n-j), j=1..n-1))/ (n-1)) end: b:= proc(n) option remember; -`if`(n<0, 1, add(b(n-i) *A81(i+1), i=1..n+1)) end: B:= proc(n) add(b(i) *x^i, i=0..n) end: T:= (n,k)-> coeff(B(n)^k, x, n-k): seq(seq(T(n, k), k=1..n), n=1..13); # Alois P. Heinz, Oct 23 2009

A81[n_] := A81[n] = If[n < 2, n, Sum[ Sum[ d*A81[d], {d, Divisors[j]} ] * A81[n-j], {j, 1, n-1}]/(n-1)]; b[n_] := b[n] = -If[n < 0, 1, Sum[ b[n-i]*A81[i+1], {i, 1, n+1}]]; B[n_] := Sum[ b[i]*x^i, {i, 0, n}]; T[n_, k_] := Coefficient[ B[n]^k, x, n-k]; Flatten[ Table[ T[n, k], {n, 1, 12}, {k, 1, n}]] (* Jean-François Alcover, Jan 20 2012, translated from Alois P. Heinz's Maple program *)

More terms from Alois P. Heinz, Oct 23 2009

A052251 Column 3 of A052250.

Original entry on oeis.org

1, 3, 6, 13, 27, 63, 148, 363, 912, 2330, 6036, 15825, 41865, 111636, 299736, 809513, 2197728, 5994219, 16416748, 45129396, 124479270, 344403494, 955557780, 2658061560, 7411457963, 20710700277, 57992124810, 162691293718, 457219737027, 1287065977413

Offset: 3

David Broadhurst, Feb 05 2000

nonn

Also expansion of cube of g.f. for A051573. - Alois P. Heinz, Oct 23 2009

Alois P. Heinz, Table of n, a(n) for n = 3..700

Cf. A051573, A000081. - Alois P. Heinz, Oct 23 2009

Cf. A051491.

with(numtheory): A81:= proc(n) option remember; `if`(n<2, n, (add(add(d*A81(d), d=divisors(j)) *A81(n-j), j=1..n-1))/ (n-1)) end: b:= proc(n) option remember; -`if`(n<0, 1, add(b(n-i) *A81(i+1), i=1..n+1)) end: B:= proc(n) add(b(i) *x^i, i=0..n) end: a:= n-> coeff(B(n)^3, x, n-3): seq(a(n), n=3..35); # Alois P. Heinz, Oct 23 2009

A81[n_] := A81[n] = If[n < 2, n, Sum[Sum[d A81[d], {d, Divisors[j]}] A81[n - j], {j, 1, n - 1}]/(n - 1)];
b[n_] := b[n] = -If[n < 0, 1, Sum[b[n - i] A81[i + 1], {i, 1, n + 1}]];
B[n_] := Sum[b[i] x^i, {i, 0, n}];
T[n_, k_] := Coefficient[B[n]^k, x, n - k];
a[n_] := T[n, 3];
a /@ Range[3, 35] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148175241..., c = 0.195489104191039910520879642... . - Vaclav Kotesovec, Sep 05 2014

More terms from Alois P. Heinz, Oct 23 2009

A052252 Column 4 of A052250.

Original entry on oeis.org

1, 4, 10, 24, 55, 132, 322, 804, 2051, 5304, 13886, 36716, 97882, 262756, 709658, 1926748, 5255707, 14396048, 39580338, 109190052, 302148814, 838449236, 2332652648, 6505071080, 18180441512, 50914047384, 142853059922, 401517522844, 1130400537667, 3187335556064

Offset: 4

David Broadhurst, Feb 05 2000

nonn

Also expansion of 4th power of g.f. for A051573. - Alois P. Heinz, Oct 23 2009

Alois P. Heinz, Table of n, a(n) for n = 4..500

Cf. A051573, A000081. - Alois P. Heinz, Oct 23 2009

with(numtheory): A81:= proc(n) option remember; `if`(n<2, n, (add(add(d*A81(d), d=divisors(j)) *A81(n-j), j=1..n-1))/ (n-1)) end: b:= proc(n) option remember; -`if`(n<0, 1, add(b(n-i) *A81(i+1), i=1..n+1)) end: B:= proc(n) add(b(i) *x^i, i=0..n) end: a:= n-> coeff(B(n)^4, x, n-4): seq(a(n), n=4..35); # Alois P. Heinz, Oct 23 2009

A81[n_] := A81[n] = If[n < 2, n, Sum[Sum[d A81[d], {d, Divisors[j]}] A81[n - j], {j, 1, n - 1}]/(n - 1)];
b[n_] := b[n] = -If[n < 0, 1, Sum[b[n - i] A81[i + 1], {i, 1, n + 1}]];
B[n_] := Sum[b[i] x^i, {i, 0, n}];
T[n_, k_] := Coefficient[B[n]^k, x, n - k];
a[n_] := T[n, 4];
a /@ Range[4, 35] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.17246782327675280347707... . - Vaclav Kotesovec, Sep 06 2014

More terms from Alois P. Heinz, Oct 23 2009

A098091 Graded dimension of the space of invariant bilinear forms on the free pre-Lie algebra on one generator.

Original entry on oeis.org

1, 1, 4, 7, 17, 36, 88, 196, 481, 1148, 2838, 7020, 17708, 44837, 114971, 296457

Offset: 5

F. Chapoton, Sep 14 2004

The sequence starts at index 5 because the previous terms vanish.

This is related in some sense to A000081.

Cf. A051573, A123256.

It seems that a(n+1) = A000081(n) - A051573(n). - Andrey Zabolotskiy, Aug 05 2024

A144963 Eigentriangle, row sums = A000081 starting (1, 2, 4, 9, 20, 48, 115, ...).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 4, 3, 2, 2, 4, 9, 8, 3, 4, 4, 9, 20, 16, 8, 6, 8, 9, 20, 48, 41, 16, 16, 12, 18, 20, 48, 115, 98, 41, 32, 32, 27, 40, 115, 286, 250, 98, 82, 64, 72, 60, 96, 115, 286, 719

Offset: 1

Gary W. Adamson, Sep 27 2008

Row sums = A000081 starting with offset 2: (1, 2, 4, 9, 20, 48, 115, ...).
Right border = (1, 1, 2, 4, 9, 20, 48, ...).
Left border = A051573: (1, 1, 1, 2, 3, 8, 16, 41, ...).
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle:
   1;
   1,  1;
   1,  1,  2;
   2,  1,  2,  4;
   3,  2,  2,  4,  9;
   8,  3,  4,  4,  9, 20;
  16,  8,  6,  8,  9, 20, 48;
  41, 16, 16, 12, 18, 20, 48, 115;
  98, 41, 32, 32, 27, 40, 48, 115, 286;
  ...
Row 4 = (2, 1, 2, 4) = termwise products of (2, 1, 1, 1) and (1, 1, 2, 4).

Eigentriangle by rows, termwise products of A000081 starting with offset 2: (1, 2, 4, 9, 20, 48, ...) and row terms of an A051573 decrescendo triangle: (1; 1,1; 1,1,1; 2,1,1,1; 3,2,1,1,1; ...) where A051573 = (1, 1, 1, 2, 3, 8, 16, 41, ...).

A115868 Invariants for a hidden action of S_(n+1) on Cayley trees with n vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 6, 11, 18, 39, 70, 153, 321, 721, 1612, 3792, 8896, 21498, 52230, 128994, 320786, 806582, 2040912, 5205311, 13352470, 34460430, 89384609

Offset: 2

F. Chapoton, Mar 14 2006

This is the multiplicity of the trivial module in a sequence of modules of dimension (n-1)^(n-3) over the symmetric groups S_n. The restriction of these modules to S_(n-1) is given by the action on trees.

			M[6]=s[2, 1, 1, 1, 1] + 3 s[2, 2, 2] + 2 s[3, 1, 1, 1] + 2 s[3, 2, 1] + s[4, 1, 1] + 4 s[4, 2] + s[5, 1] + 2 s[6] as a sum of Schur functions hence a[6]=2.

Cf. A000055 and A000272.

Cf. A000081, A051573, A098091.

No simple formula known, only a complicated sum over partitions.

It seems that a(n+1) = A000055(n) + A051573(n) - A000081(n). - Andrey Zabolotskiy, Aug 05 2024

Five more terms added by F. Chapoton, Mar 08 2020

cp's OEIS Frontend

A051603 Expansion of square of g.f. for A051573.

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A052250 Triangle T(n,k) (n >= 1, k >= 1) giving dimension of bigrading of Hopf algebra of rooted trees.

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A052251 Column 3 of A052250.

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A052252 Column 4 of A052250.

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A098091 Graded dimension of the space of invariant bilinear forms on the free pre-Lie algebra on one generator.

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A144963 Eigentriangle, row sums = A000081 starting (1, 2, 4, 9, 20, 48, 115, ...).

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A115868 Invariants for a hidden action of S_(n+1) on Cayley trees with n vertices.

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Extensions