A292462 -id:A292462 - cp's OEIS frontend

A292463 Number of partitions of n with n kinds of 1.

1, 1, 4, 14, 51, 188, 702, 2644, 10026, 38223, 146359, 562456, 2168134, 8379539, 32459199, 125984039, 489837300, 1907490728, 7438346255, 29042470132, 113522618066, 444199913556, 1739735079466, 6819657196928, 26753893533257, 105034060120469, 412637434996367

Offset: 0

Alois P. Heinz, Sep 16 2017

nonn

			a(2) = 4: 2, 1a1a, 1a1b, 1b1b.

Alois P. Heinz, Table of n, a(n) for n = 0..1664

Main diagonal of A292508.

Cf. A000041, A014329, A292462, A292503, A292507, A292541.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+k-1)*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=1,
      combinat[numbpart](n), b(n-1, k) +b(n, k-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

Table[SeriesCoefficient[1/(1-x)^(n-1) * Product[1/(1-x^k), {k,1,n}], {x,0,n}], {n,0,30}] (* Vaclav Kotesovec, Sep 19 2017 *)

a(n) = [x^n] 1/(1-x)^n * 1/Product_{j=2..n} (1-x^j).
a(n) is n-th term of the Euler transform of n,1,1,1,... .
a(n) ~ c * 4^n / sqrt(n), where c = QPochhammer[-1, 1/2] / (8*sqrt(Pi) * QPochhammer[1/4, 1/4]) = 0.48841139329043831428669851139824427133317... - Vaclav Kotesovec, Sep 19 2017
Equivalently, c = 1/(4*sqrt(Pi)*QPochhammer(1/2)). - Vaclav Kotesovec, Mar 17 2024

A292503 Number of partitions of n with n sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 1, 3, 7, 20, 63, 233, 966, 4454, 22404, 121616, 706362, 4361977, 28494493, 196087988, 1416515642, 10709058487, 84505818259, 694397612486, 5929368380664, 52513737017847, 481577858196052, 4565851595293151, 44692014464166068, 451058715629365617

Offset: 0

Alois P. Heinz, Sep 17 2017

nonn

			a(2) = 3: 2, 1a1a, 1a1b.
a(3) = 7: 3, 21a, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1a1b1c.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A292462, A292463, A292507.
Main diagonal of A292745.
Row sums of A292746.

f:= (n, k)-> add(Stirling2(n, j), j=0..k):
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..30);

f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, f[n, k], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

A292741 Number A(n,k) of partitions of n with k sorts of part 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 2, 1, 4, 10, 11, 5, 2, 1, 5, 17, 31, 24, 7, 4, 1, 6, 26, 69, 95, 50, 11, 4, 1, 7, 37, 131, 278, 287, 104, 15, 7, 1, 8, 50, 223, 657, 1114, 865, 212, 22, 8, 1, 9, 65, 351, 1340, 3287, 4460, 2599, 431, 30, 12, 1, 10, 82, 521, 2459, 8042, 16439, 17844, 7804, 870, 42, 14

Offset: 0

Alois P. Heinz, Sep 22 2017

			A(1,3) = 3: 1a, 1b, 1c.
A(2,3) = 10: 2, 1a1a, 1a1b, 1a1c, 1b1a, 1b1b, 1b1c, 1c1a, 1c1b, 1c1c.
A(3,2) = 11: 3, 21a, 21b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,     1,      1,      1, ...
  0,  1,   2,    3,     4,     5,      6,      7, ...
  1,  2,   5,   10,    17,    26,     37,     50, ...
  1,  3,  11,   31,    69,   131,    223,    351, ...
  2,  5,  24,   95,   278,   657,   1340,   2459, ...
  2,  7,  50,  287,  1114,  3287,   8042,  17215, ...
  4, 11, 104,  865,  4460, 16439,  48256, 120509, ...
  4, 15, 212, 2599, 17844, 82199, 289540, 843567, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-2 give: A002865, A000041, A090764.
Rows n=0-2 give: A000012, A001477, A002522, A071568.
Main diagonal gives A292462.
Cf. A003992, A004248, A009998, A051129, A292508, A292622, A292745.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, k^n,
      add(b(n-i*j, i-1, k), j=0..iquo(n, i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[0, , ] = 1; b[n_, i_, k_] := b[n, i, k] = If[i < 2, k^n, Sum[b[n - i*j, i - 1, k], {j, 0, Quotient[n, i]}]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 19 2018, translated from Maple *)

G.f. of column k: 1/(1-k*x) * 1/Product_{j>=2} (1-x^j).

A(n,k) = Sum_{j=0..n} A002865(j) * k^(n-j).

A292507 Number of partitions of n with up to n distinct kinds of 1.

Original entry on oeis.org

1, 1, 2, 5, 13, 33, 82, 201, 488, 1176, 2817, 6714, 15931, 37647, 88628, 207914, 486158, 1133304, 2634339, 6106953, 14121157, 32573842, 74968044, 172164086, 394561089, 902471184, 2060338222, 4695324425, 10681885697, 24261437446, 55017434305, 124573678280

Offset: 0

Alois P. Heinz, Sep 17 2017

nonn

			a(3) = 5: 3, 21a, 21b, 21c, 1a1b1c.
a(4) = 13: 4, 31a, 31b, 31c, 31d, 22, 21a1b, 21a1c, 21a1d, 21b1c, 21b1d, 21c1d, 1a1b1c1d.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1382 (terms 0..1000 from Alois P. Heinz)

Main diagonal of A292622.

Cf. A292462, A292463, A292503.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1,
      binomial(k, n), `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..35);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, Binomial[k, n], If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

Conjecture: log(a(n)) ~ log(2)*n + Pi*sqrt(n/3) - 3*log(n)/2. - Vaclav Kotesovec, May 11 2019

a(n) = [x^n] (1 + x)^n * Product_{k>=2} 1 / (1 - x^k). - Ilya Gutkovskiy, Apr 24 2021

A292567 a(n) = [x^n] 1/(1+nx) Product_{j=2..n} 1/(1-x^j).

Original entry on oeis.org

1, -1, 5, -29, 270, -3233, 47800, -838561, 17013991, -391779640, 10091836632, -287491284748, 8973657413421, -304549220113387, 11165193890312790, -439726629957500944, 18514829984975265703, -829953080825411342745, 39461813340364709540008

Offset: 0

Alois P. Heinz, Sep 19 2017

sign

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A002865, A292462.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, (-k)^n,
      `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..23);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, (-k)^n, If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];
a[n_] := If[n == 0, 1, b[n, n, n]];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)

a(n) ~ (-1)^n * n^n * (1 + 1/n^2 - 1/n^3 + 2/n^4 - 2/n^5 + 4/n^6 - 4/n^7 + 7/n^8 - 8/n^9 + 12/n^10), for coefficients see A002865. - Vaclav Kotesovec, Sep 19 2017

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A292463 Number of partitions of n with n kinds of 1.

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A292503 Number of partitions of n with n sorts of part 1 which are introduced in ascending order.

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A292741 Number A(n,k) of partitions of n with k sorts of part 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A292507 Number of partitions of n with up to n distinct kinds of 1.

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A292567 a(n) = [x^n] 1/(1+n*x) * Product_{j=2..n} 1/(1-x^j).

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A292567 a(n) = [x^n] 1/(1+nx) Product_{j=2..n} 1/(1-x^j).