A292746 -id:A292746 - cp's OEIS frontend

A292745 Number A(n,k) of partitions of n with k sorts of part 1 which are introduced in ascending order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 3, 6, 5, 2, 1, 1, 3, 7, 13, 7, 4, 1, 1, 3, 7, 19, 26, 11, 4, 1, 1, 3, 7, 20, 52, 54, 15, 7, 1, 1, 3, 7, 20, 62, 151, 108, 22, 8, 1, 1, 3, 7, 20, 63, 217, 442, 219, 30, 12, 1, 1, 3, 7, 20, 63, 232, 803, 1314, 439, 42, 14

Offset: 0

Alois P. Heinz, Sep 22 2017

			A(3,2) = 6: 3, 21a, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,    1,    1,    1,    1,    1, ...
  0,  1,   1,    1,    1,    1,    1,    1,    1, ...
  1,  2,   3,    3,    3,    3,    3,    3,    3, ...
  1,  3,   6,    7,    7,    7,    7,    7,    7, ...
  2,  5,  13,   19,   20,   20,   20,   20,   20, ...
  2,  7,  26,   52,   62,   63,   63,   63,   63, ...
  4, 11,  54,  151,  217,  232,  233,  233,  233, ...
  4, 15, 108,  442,  803,  944,  965,  966,  966, ...
  7, 22, 219, 1314, 3092, 4158, 4425, 4453, 4454, ...

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

Columns k=0-10 give: A002865, A000041, A320733, A320734, A320735, A320736, A320737, A320738, A320739, A320740, A320741.
Main diagonal gives A292503.
Cf. A292508, A292622, A292741, 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, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

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_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 17 2018, translated from Maple *)

A(n,k) = Sum_{j=0..k} A292746(n,j).

A(n,k) = A(n,n) for all k >= n.

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 *)

A259401 a(n) = Sum_{k=0..n} 2^(n-k)*p(k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 3, 8, 19, 43, 93, 197, 409, 840, 1710, 3462, 6980, 14037, 28175, 56485, 113146, 226523, 453343, 907071, 1814632, 3629891, 7260574, 14522150, 29045555, 58092685, 116187328, 232377092, 464757194, 929518106, 1859040777, 3718087158, 7436181158, 14872370665

Offset: 0

Vaclav Kotesovec, Jun 26 2015

nonn

In general, Sum_{k=0..n} (m^(n-k) * p(k)) ~ m^n / QPochhammer[1/m, 1/m], for m > 1.

Alois P. Heinz, Table of n, a(n) for n = 0..3320

Cf. A000041, A048651, A090764, A259400, A292746.

a:= proc(n) option remember; `if`(n<0, 0,
      2*a(n-1)+combinat[numbpart](n))
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 03 2019

Table[Sum[2^(n-k)*PartitionsP[k],{k,0,n}],{n,0,50}]

a(n) = sum(k=0, n, 2^(n-k)*numbpart(k)); \\ Michel Marcus, Dec 03 2019

a(n) ~ c * 2^n, where c = 1/A048651 = 1/QPochhammer[1/2, 1/2] = 3.462746619455...

G.f.: (1/(1 - 2*x)) * Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Dec 03 2019

A292747 Number of partitions of 2n with exactly n kinds of 1's which are introduced in ascending order.

Original entry on oeis.org

1, 1, 8, 97, 1778, 43747, 1349703, 50033463, 2164920950, 107074391802, 5957871478583, 368330684797595, 25046735249606820, 1857906353180702199, 149289720057575358424, 12917953683720554797237, 1197556745092101849164899, 118414507831659267311128558

Offset: 0

Alois P. Heinz, Sep 22 2017

nonn

			a(2) = 8: 21a1b, 1a1a1a1b, 1a1a1b1a, 1a1a1b1b, 1a1b1a1a, 1a1b1a1b, 1a1b1b1a, 1a1b1b1b  (the two kinds of 1's are denoted by 1a and 1b).

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. 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(2*n$2, n)-b(2*n$2, n-1):
seq(a(n), n=0..20);

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[2n, 2n, n] - b[2n, 2n, n-1];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)

a(n) = A292746(2n,n).

a(n) ~ 2^(2*n) * n^(n-1/2) / (sqrt(2*Pi*(1-c)) * exp(n) * c^n * (2-c)^n), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.40637573995995990767695812412483975821... - Vaclav Kotesovec, Sep 28 2017

A320816 Number of partitions of n with exactly three sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 6, 26, 97, 334, 1095, 3482, 10855, 33405, 101925, 309237, 934691, 2818110, 8482505, 25504000, 76625146, 230101961, 690759226, 2073184749, 6221368879, 18667736528, 56010470158, 168045932624, 504166843427, 1512558622966, 4537792056226, 13613608545770

Offset: 3

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..2097

Column k=3 of A292746.

Cf. A320733, A320734.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(3):
seq(a(n), n=3..35);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, k}], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
a[n_] := With[{k = 3}, b[n, n, k] - b[n, n, k - 1]];
a /@ Range[3, 35] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

a(n) = A320734(n) - A320733(n).

A320817 Number of partitions of n with exactly four sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 10, 66, 361, 1778, 8207, 36310, 156095, 657785, 2733065, 11241497, 45900679, 186420826, 754165809, 3042167236, 12245294090, 49211278321, 197535872510, 792216674789, 3175088068035, 12719020008668, 50932090504830, 203896407951944, 816089798651203

Offset: 4

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1663

Column k=4 of A292746.

Cf. A320734, A320735.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(4):
seq(a(n), n=4..35);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, k}], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
a[n_] := With[{k = 4}, b[n, n, k] - b[n, n, k-1]];
a /@ Range[4, 35] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

a(n) = A320735(n) - A320734(n).

A320818 Number of partitions of n with exactly five sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 15, 141, 1066, 7108, 43747, 255045, 1431320, 7814385, 41804990, 220266447, 1147232914, 5922585396, 30367092789, 154877631181, 786633449995, 3982378528296, 20109428513990, 101339359244739, 509871884291730, 2562078441467318, 12861324297841420

Offset: 5

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1433

Column k=5 of A292746.

Cf. A320735, A320736.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(5):
seq(a(n), n=5..35);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, k}], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
a[n_] := With[{k = 5}, b[n, n, k] - b[n, n, k - 1]];
a /@ Range[5, 35] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

a(n) = A320736(n) - A320735(n).

A320819 Number of partitions of n with exactly six sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 21, 267, 2668, 23116, 182443, 1349703, 9529538, 64991613, 431754668, 2810794455, 18011999644, 113994583260, 714334592349, 4440885185275, 27431944561645, 168574045898166, 1031553703902986, 6290661582662655, 38253841380267660, 232085126723073278

Offset: 6

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1288

Column k=6 of A292746.

Cf. A320736, A320737.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(6):
seq(a(n), n=6..35);

a(n) = A320737(n) - A320736(n).

A320820 Number of partitions of n with exactly seven sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 28, 463, 5909, 64479, 633796, 5786275, 50033463, 415225854, 3338335646, 26179143977, 201266007483, 1522856635641, 11374331041836, 84061202478127, 615860361908534, 4479596579257904, 32388729758708314, 233011769893620853, 1669336230635613631

Offset: 7

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1187

Column k=7 of A292746.

Cf. A320737, A320738.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(7):
seq(a(n), n=7..35);

a(n) = A320738(n) - A320737(n).

A320821 Number of partitions of n with exactly eight sorts of part 1 which are introduced in ascending order.

Original entry on oeis.org

1, 36, 751, 11917, 159815, 1912316, 21084803, 218711887, 2164920950, 20657703246, 191440769945, 1732792167043, 15385193971985, 134455882817716, 1159708265019855, 9893526482067374, 83627808435796896, 701411197245083482, 5844301347854288709, 48423747013469923303

Offset: 8

Alois P. Heinz, Oct 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1112

Column k=8 of A292746.

Cf. A320738, A320739.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(8):
seq(a(n), n=8..35);

a(n) = A320739(n) - A320738(n).

cp's OEIS Frontend

A292745 Number A(n,k) of partitions of n with k sorts of part 1 which are introduced in ascending order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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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A259401 a(n) = Sum_{k=0..n} 2^(n-k)*p(k), where p(k) is the partition function A000041.

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A292747 Number of partitions of 2n with exactly n kinds of 1's which are introduced in ascending order.

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A320816 Number of partitions of n with exactly three sorts of part 1 which are introduced in ascending order.

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A320817 Number of partitions of n with exactly four sorts of part 1 which are introduced in ascending order.

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A320818 Number of partitions of n with exactly five sorts of part 1 which are introduced in ascending order.

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A320819 Number of partitions of n with exactly six sorts of part 1 which are introduced in ascending order.

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