A294746 -id:A294746 - cp's OEIS frontend

A294857 Number of compositions (ordered partitions) of 1 into exactly 9*n+1 powers of 1/10.

1, 1, 92378, 638048716305, 79376343363731999772, 87451065686506297464527100009, 556054452693724489326948624520266970011, 15314213846343665104211951568419481838710411193512, 1486749321998795158270268184508036244709339593298119460028356

Offset: 0

Alois P. Heinz, Nov 09 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..64

Column k=9 of A294746.

a(n) = [x^(10^n)] (Sum_{j=0..9*n+1} x^(10^j))^(9*n+1).

A294858 Number of compositions (ordered partitions) of 1 into exactly 10*n+1 powers of 1/11.

Original entry on oeis.org

1, 1, 352716, 15643738390215, 17538231051073300512165, 224099040671253218432160498959100, 20208669423838553069878798723999482271266772, 9331135718988942028135354112509213417271737533460581101, 17527579661734914837260482461964964314280710461958157215893286079640

Offset: 0

Alois P. Heinz, Nov 09 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..59

Column k=10 of A294746.

a(n) = [x^(11^n)] (Sum_{j=0..10*n+1} x^(11^j))^(10*n+1).

A294982 Number of compositions (ordered partitions) of 1 into exactly 3n+1 powers of 1/(n+1).

Original entry on oeis.org

1, 13, 217, 4245, 90376, 2019836, 46570140, 1097525253, 26293568950, 638048716305, 15643738390215, 386826618273420, 9633468179090952, 241366000080757480, 6078975012187601768, 153798067122829610085, 3906583987216447704594, 99579591801208823965115

Offset: 0

Alois P. Heinz, Nov 12 2017

nonn

			a(0) = 1: [1].
a(1) = 13: [1/4,1/4,1/4,1/4], [1/2,1/4,1/8,1/8], [1/2,1/8,1/4,1/8], [1/2,1/8,1/8,1/4], [1/4,1/2,1/8,1/8], [1/4,1/8,1/2,1/8], [1/4,1/8,1/8,1/2], [1/8,1/2,1/4,1/8], [1/8,1/2,1/8,1/4], [1/8,1/4,1/2,1/8], [1/8,1/4,1/8,1/2], [1/8,1/8,1/2,1/4], [1/8,1/8,1/4,1/2].

Alois P. Heinz, Table of n, a(n) for n = 0..700

Row n=3 of A294746.

a:= proc(n) option remember; `if`(n<2, 12*n+1, (3*n-1)*(3*n+1)*
      3*((15*n^3-31*n^2-4*n+8)*n*a(n-1)-3*(3*n-4)*(3*n-2)*
      (3*n^2-2*n-2)*a(n-2))/((n+1)*(4*n+2)*(3*n^2-8*n+3)*n^2))
    end:
seq(a(n), n=0..20);

a[n_] := a[n] = If[n < 2, 12*n + 1, (3*n - 1)*(3*n + 1)*3*((15*n^3 - 31*n^2 - 4*n + 8)*n*a[n-1] - 3*(3*n - 4)*(3*n - 2)*(3*n^2 - 2*n - 2)*a[n-2])/((n + 1)*(4*n + 2)*(3*n^2 - 8*n + 3)*n^2)];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 21 2018, translated from Maple *)
Table[Binomial[3*n + 1, n + 1]*Binomial[2*n, n] + Binomial[3*n + 1, n - 1], {n, 0, 20}] (* Vaclav Kotesovec, Sep 20 2019 *)

For n>0, a(n) = binomial(3*n+1, n+1)*binomial(2*n, n) + binomial(3*n+1, n-1). - Vaclav Kotesovec, Sep 20 2019

A294983 Number of compositions (ordered partitions) of 1 into exactly 4n+1 powers of 1/(n+1).

Original entry on oeis.org

1, 75, 8317, 1239823, 216456376, 41175714454, 8251690444250, 1713228373452375, 365077361327242168, 79376343363731999772, 17538231051073300512165, 3926523351382832339690135, 888819911396229761050640552, 203083664214425241278951079860

Offset: 0

Alois P. Heinz, Nov 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..416

Row n=4 of A294746.

b:= proc(n, r, p, k) option remember;
      `if`(n (k-> `if`(n=0, 1, b(k*n+1, 1, 0, n+1)))(4):
seq(a(n), n=0..20);

b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
a[n_] := If[n == 0, 1, b[#*n + 1, 1, 0, n + 1]]&[4];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

From Vaclav Kotesovec, Sep 20 2019: (Start)
Recurrence: 3*(n-1)^2*n^3*(n+1)*(2*n + 1)*(3*n + 1)*(3*n + 2)*(1794*n^8 - 24743*n^7 + 137870*n^6 - 403038*n^5 + 671286*n^4 - 647019*n^3 + 349914*n^2 - 97024*n + 10624)*a(n) = 4*(n-1)^2*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*(249366*n^11 - 3033833*n^10 + 13899420*n^9 - 28885850*n^8 + 21978374*n^7 + 11319871*n^6 - 24589048*n^5 + 5121972*n^4 + 6329552*n^3 - 2255040*n^2 - 432000*n + 176256)*a(n-1) - 64*(n-1)*(2*n - 3)*(2*n - 1)*(4*n - 5)*(4*n - 3)*(4*n - 1)*(4*n + 1)*(28704*n^10 - 307085*n^9 + 1172466*n^8 - 1799547*n^7 + 417441*n^6 + 1564368*n^5 - 1050675*n^4 - 368572*n^3 + 336852*n^2 + 24624*n - 25056)*a(n-2) + 256*(2*n - 5)*(2*n - 3)*(2*n - 1)*(4*n - 9)*(4*n - 7)*(4*n - 5)*(4*n - 3)*(4*n - 1)*(4*n + 1)*(1794*n^8 - 10391*n^7 + 14901*n^6 + 5043*n^5 - 16279*n^4 - 396*n^3 + 4872*n^2 + 72*n - 336)*a(n-3).
a(n) ~ 2^(8*n + 3/2) / (Pi^(3/2) * n^(3/2)). (End)

A294984 Number of compositions (ordered partitions) of 1 into exactly 5n+1 powers of 1/(n+1).

Original entry on oeis.org

1, 525, 487630, 709097481, 1303699790001, 2713420774885145, 6078597035484932995, 14303426764164190428105, 34883776613634643730481238, 87451065686506297464527100009, 224099040671253218432160498959100, 584668421756097333754383886118155275

Offset: 0

Alois P. Heinz, Nov 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..287

Row n=5 of A294746.

b:= proc(n, r, p, k) option remember;
      `if`(n (k-> `if`(n=0, 1, b(k*n+1, 1, 0, n+1)))(5):
seq(a(n), n=0..15);

b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
a[n_] := If[n == 0, 1, b[#*n + 1, 1, 0, n + 1]]&[5];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) ~ 5^(5*n + 3/2) / (4 * Pi^2 * n^2). - Vaclav Kotesovec, Sep 20 2019

A294985 Number of compositions (ordered partitions) of 1 into exactly 6n+1 powers of 1/(n+1).

Original entry on oeis.org

1, 4347, 40647178, 701954099115, 16596702491586251, 461871979542736134676, 14138484434475011392912026, 460977928965130046448503507051, 15732393344641740454307566725567376, 556054452693724489326948624520266970011, 20208669423838553069878798723999482271266772

Offset: 0

Alois P. Heinz, Nov 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..215

Row n=6 of A294746.

b:= proc(n, r, p, k) option remember;
      `if`(n (k-> `if`(n=0, 1, b(k*n+1, 1, 0, n+1)))(6):
seq(a(n), n=0..15);

b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
a[n_] := If[n == 0, 1, b[#*n + 1, 1, 0, n + 1]]&[6];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) ~ 6^(6*n + 3/2) / (2*Pi*n)^(5/2). - Vaclav Kotesovec, Sep 20 2019

A294986 Number of compositions (ordered partitions) of 1 into exactly 7n+1 powers of 1/(n+1).

Original entry on oeis.org

1, 41245, 4561368175, 1104353764428365, 396695587555058190126, 174436242482643190451211853, 86237678200608256132213084584295, 46050764886573707269872023694736134925, 25997337847684377365651388718120083246723460

Offset: 0

Alois P. Heinz, Nov 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..170

Row n=7 of A294746.

b:= proc(n, r, p, k) option remember;
      `if`(n (k-> `if`(n=0, 1, b(k*n+1, 1, 0, n+1)))(7):
seq(a(n), n=0..12);

b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
a[n_] := If[n == 0, 1, b[#*n + 1, 1, 0, n + 1]]&[7];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) ~ 7^(7*n + 3/2) / (8 * Pi^3 * n^3). - Vaclav Kotesovec, Sep 20 2019

A294987 Number of compositions (ordered partitions) of 1 into exactly 8n+1 powers of 1/(n+1).

Original entry on oeis.org

1, 441675, 663134389930, 2594884910993019575, 16336038155342083651640376, 130958058407369286623026190867082, 1206534243283932582765850205674424343577, 12176825528022093702548525617184407475359333407, 131223281654667714701311635640432890136981994039662720

Offset: 0

Alois P. Heinz, Nov 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..139

Row n=8 of A294746.

b:= proc(n, r, p, k) option remember;
      `if`(n (k-> `if`(n=0, 1, b(k*n+1, 1, 0, n+1)))(8):
seq(a(n), n=0..12);

b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
a[n_] := If[n == 0, 1, b[#*n + 1, 1, 0, n + 1]]&[8];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) ~ 8^(8*n + 3/2) / (2*Pi*n)^(7/2). - Vaclav Kotesovec, Sep 20 2019

A294988 Number of compositions (ordered partitions) of 1 into exactly 9n+1 powers of 1/(n+1).

Original entry on oeis.org

1, 5259885, 121218250616173, 8684483842898500680225, 1085776473843765315524916060126, 179835209135492330050411858875313971595, 34994508245963099403565066291175900528344592700, 7565469782615095731665958935875509379368611893407583633

Offset: 0

Alois P. Heinz, Nov 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..117

Row n=9 of A294746.

b:= proc(n, r, p, k) option remember;
      `if`(n (k-> `if`(n=0, 1, b(k*n+1, 1, 0, n+1)))(9):
seq(a(n), n=0..10);

b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
a[n_] := If[n == 0, 1, b[#*n + 1, 1, 0, n + 1]]&[9];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) ~ 9^(9*n + 3/2) / (16 * Pi^4 * n^4). - Vaclav Kotesovec, Sep 20 2019

A294989 Number of compositions (ordered partitions) of 1 into exactly 10n+1 powers of 1/(n+1).

Original entry on oeis.org

1, 68958747, 27212315953140892, 39880061006390454401626995, 110656003660578876500875377620844376, 423205992807070499591372608204571223421862945, 1944053748730350081768848916806347783770184147756976500

Offset: 0

Alois P. Heinz, Nov 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..101

Row n=10 of A294746.

b:= proc(n, r, p, k) option remember;
      `if`(n (k-> `if`(n=0, 1, b(k*n+1, 1, 0, n+1)))(10):
seq(a(n), n=0..10);

b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
a[n_] := If[n == 0, 1, b[#*n + 1, 1, 0, n + 1]]&[10];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) ~ 10^(10*n + 3/2) / (2*Pi*n)^(9/2). - Vaclav Kotesovec, Sep 20 2019

cp's OEIS Frontend

A294857 Number of compositions (ordered partitions) of 1 into exactly 9*n+1 powers of 1/10.

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A294858 Number of compositions (ordered partitions) of 1 into exactly 10*n+1 powers of 1/11.

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A294982 Number of compositions (ordered partitions) of 1 into exactly 3n+1 powers of 1/(n+1).

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A294983 Number of compositions (ordered partitions) of 1 into exactly 4n+1 powers of 1/(n+1).

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Maple

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A294984 Number of compositions (ordered partitions) of 1 into exactly 5n+1 powers of 1/(n+1).

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Maple

Mathematica

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A294985 Number of compositions (ordered partitions) of 1 into exactly 6n+1 powers of 1/(n+1).

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Maple

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A294986 Number of compositions (ordered partitions) of 1 into exactly 7n+1 powers of 1/(n+1).

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Maple

Mathematica

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A294987 Number of compositions (ordered partitions) of 1 into exactly 8n+1 powers of 1/(n+1).

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Programs

Maple

Mathematica

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A294988 Number of compositions (ordered partitions) of 1 into exactly 9n+1 powers of 1/(n+1).

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