A052250 -id:A052250 - cp's OEIS frontend

A052251 Column 3 of A052250.

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

A051573 INVERTi transform of A000081 = [1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486,...].

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 16, 41, 98, 250, 631, 1646, 4285, 11338, 30135, 80791, 217673, 590010, 1606188, 4392219, 12055393, 33206321, 91752211, 254261363, 706465999, 1967743066, 5493195530, 15367129299, 43073007846, 120949992543, 340206026166, 958444631917

Offset: 0

N. J. A. Sloane

nonn

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

Cf. A052250, A000081.

with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<2, n,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
a:= proc(n) option remember; local i; `if`(n<0, -1,
      -add(a(n-i) *b(i+1), i=1..n+1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 17 2013

b[n_] := b[n] = If[n < 2, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)]; a[n_] := a[n] = If[n < 0, -1, -Sum[a[n-i]*b[i+1], {i, 1, n+1}]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 16 2014, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148175241..., c = A187770 = 0.4399240125710253040409033914... . - Vaclav Kotesovec, Sep 06 2014

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

Original entry on oeis.org

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 *)

A052249 Triangle T(n,k) (n >= 1, k >= 1) giving dimension of bigrading of Connes-Moscovici noncocommutative algebra.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 2, 4, 1, 0, 0, 1, 4, 5, 1, 0, 0, 0, 2, 6, 6, 1, 0, 0, 0, 1, 4, 9, 7, 1, 0, 0, 0, 0, 2, 7, 12, 8, 1, 0, 0, 0, 0, 1, 4, 11, 16, 9, 1, 0, 0, 0, 0, 0, 2, 7, 16, 20, 10, 1, 0, 0, 0, 0, 0, 1, 4, 12, 23, 25, 11, 1, 0, 0, 0, 0, 0, 0, 2, 7, 18, 31, 30, 12, 1, 0, 0

Offset: 0

David Broadhurst, Feb 05 2000

With rows reversed, T(n,k) appears to be the number of partitions of n with k big parts, where a big part is a part >= 2 (0 <= k <= n/2). For example, with n=4, the 3 partitions 4, 31, 211 each have one big part. - David Callan, Aug 23 2011

			Triangle begins
  1;
  1, 1;
  0, 2, 1;
  0, 1, 3, 1;
  0, 0, 2, 4, 1;
  0, 0, 1, 4, 5, 1;
  ...

D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees, arXiv:hep-th/0001202, 2000.

Cf. A052250.

t[n_, k_] := Count[ IntegerPartitions[n], pp_ /; Count[pp, p_ /; p >= 2] == k]; Flatten[ Table[ t[n, k], {n, 1, 14}, {k, n-1, 0, -1} ] ] (* Jean-François Alcover, Jan 23 2012, after David Callan *)

A052253 Triangle T(n,k) (n >= 1, k >= 1) giving filtration of bigrading of Hopf algebra of rooted trees.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 8, 5, 4, 2, 1, 16, 13, 9, 6, 3, 1, 41, 28, 21, 13, 8, 3, 1, 98, 71, 49, 33, 20, 10, 4, 1, 250, 174, 121, 79, 50, 27, 13, 4, 1, 631, 445, 304, 201, 127, 74, 38, 16, 5, 1, 1646, 1137, 776, 510, 325, 192, 106, 49, 19, 5, 1, 4285, 2974

Offset: 0

David Broadhurst, Feb 05 2000

			1; 1,1; 2,1,1; 3,3,2,1; 8,5,4,2,1; ...

D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization...

Cf. A052250.

cp's OEIS Frontend

A052251 Column 3 of A052250.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A052252 Column 4 of A052250.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A051573 INVERTi transform of A000081 = [1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486,...].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A052249 Triangle T(n,k) (n >= 1, k >= 1) giving dimension of bigrading of Connes-Moscovici noncocommutative algebra.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A052253 Triangle T(n,k) (n >= 1, k >= 1) giving filtration of bigrading of Hopf algebra of rooted trees.

Views

Author

Keywords

Examples

Links

Crossrefs