A341222 -id:A341222 - cp's OEIS frontend

A341243 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^4.

1, 0, 4, 4, 10, 16, 26, 44, 63, 100, 144, 212, 297, 420, 584, 796, 1081, 1452, 1940, 2556, 3355, 4372, 5668, 7288, 9327, 11892, 15076, 19012, 23884, 29904, 37276, 46284, 57276, 70680, 86918, 106528, 130220, 158784, 193054, 234076, 283178, 341824, 411616, 494512, 592933

Offset: 4

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Cf. A000700, A001482, A022599, A112160, A327382, A338463, A341222, A341241, A341244, A341245, A341246, A341247, A341251.

Column k=4 of A341279.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..48);  # Alois P. Heinz, Feb 07 2021

nmax = 48; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^4.

a(n) ~ A112160(n). - Vaclav Kotesovec, Feb 20 2021

A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 22, 21, 8, 1, 11, 43, 59, 36, 10, 1, 15, 80, 144, 124, 55, 12, 1, 22, 141, 321, 362, 225, 78, 14, 1, 30, 240, 669, 944, 765, 370, 105, 16, 1, 42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1, 56, 640, 2511, 5100, 6215, 4848, 2478, 824, 171, 20, 1

Offset: 1

Alford Arnold, Apr 16 2001

Also the convolution triangle of A000041. - Peter Luschny, Oct 07 2022

			Table begins:
   1;
   2,   1;
   3,   4,    1;
   5,  10,    6,    1;
   7,  22,   21,    8,    1;
  11,  43,   59,   36,   10,    1;
  15,  80,  144,  124,   55,   12,   1;
  22, 141,  321,  362,  225,   78,  14,   1;
  30, 240,  669,  944,  765,  370, 105,  16,  1;
  42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1;
  ...
For n=4 there are 5 partitions of 4, namely 4, 31, 22, 211, 11111. There are 5 ways to pick 1 of them; 10 ways to partition one of them into 2 ordered parts: 3,1; 1,3; 2,2; 21,1; 1,21; 2,11; 11,2; 111,1; 1,111; 11,11; 6 ways to partition one of them into 3 ordered parts: 2,1,1; 1,2,1; 1,1,2; 11,1,1; 1,11,1; 1,1,11; and one way to partition one of them into 4 ordered parts: 1,1,1,1. So row 4 is 5,10,6,1.

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=1-10 give: A000041, A048574, A341221, A341222, A341223, A341225, A341226, A341227, A341228, A341236.
Row sums give A055887.
Cf. A097805, A010815, A055888, A144064, A261719, A326346.
T(2n,n) gives A340987.

A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      A(n-j, k)*numtheory[sigma](j), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 12 2015
# Uses function PMatrix from A357368. Adds row and column for n, k = 0.
PMatrix(10, combinat:-numbpart); # Peter Luschny, Oct 07 2022

A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[1, j], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[ Table[ T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} A000041(n-k)*A(k;x)*x, A(0;x) = 1. - Vladeta Jovovic, Jan 02 2004
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144064(n,k-i). - Alois P. Heinz, Mar 12 2015
Sum_{k=1..n} k * T(n,k) = A326346(n). - Alois P. Heinz, Sep 11 2019
Sum_{k=0..n} (-1)^k * T(n,k) = A010815(n). - Alois P. Heinz, Feb 07 2021
G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j))^k. - Ilya Gutkovskiy, Feb 13 2021

More terms from Vladeta Jovovic, Jan 02 2004

A341221 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^3.

Original entry on oeis.org

1, 6, 21, 59, 144, 321, 669, 1323, 2511, 4604, 8202, 14253, 24241, 40449, 66363, 107234, 170910, 269004, 418566, 644436, 982536, 1484482, 2223942, 3305484, 4876620, 7144455, 10398123, 15039564, 21624678, 30919323, 43973708, 62222844, 87619212, 122810585

Offset: 3

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Column k=3 of A060642.

Cf. A000041, A000716, A048574, A327381, A341222, A341223, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..36);  # Alois P. Heinz, Feb 07 2021

nmax = 36; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

a(n) ~ A000716(n). - Vaclav Kotesovec, Feb 20 2021

A341223 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^5.

Original entry on oeis.org

1, 10, 55, 225, 765, 2287, 6215, 15680, 37265, 84300, 182933, 383070, 777705, 1536490, 2963120, 5592060, 10349465, 18817760, 33665870, 59341785, 103176877, 177131330, 300530125, 504318530, 837632700, 1377874861, 2246061540, 3630059510, 5819556060, 9258393655, 14622472250

Offset: 5

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Column k=5 of A060642.

Cf. A000041, A023004, A048574, A327383, A341221, A341222, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..35);  # Alois P. Heinz, Feb 07 2021

nmax = 35; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

A341225 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^6.

Original entry on oeis.org

1, 12, 78, 370, 1437, 4848, 14719, 41148, 107610, 266296, 628941, 1427118, 3127369, 6646440, 13746081, 27744926, 54782271, 106029918, 201512970, 376630680, 693161334, 1257641676, 2251764699, 3982196910, 6961522279, 12038699766, 20607718317, 34938910360

Offset: 6

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..10000

Column k=6 of A060642.

Cf. A000041, A023005, A048574, A327384, A341221, A341222, A341223, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..33);  # Alois P. Heinz, Feb 07 2021

nmax = 33; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &

A341226 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^7.

Original entry on oeis.org

1, 14, 105, 567, 2478, 9317, 31269, 95965, 273896, 735966, 1879059, 4591342, 10797290, 24549924, 54171729, 116368308, 243991034, 500446135, 1006039762, 1985480063, 3852429483, 7358212272, 13850448185, 25718189483, 47150564517, 85417834621, 153015826880

Offset: 7

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..10000

Column k=7 of A060642.

Cf. A000041, A023006, A048574, A327385, A341221, A341222, A341223, A341225, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..33);  # Alois P. Heinz, Feb 07 2021

nmax = 33; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^7, {x, 0, nmax}], x] // Drop[#, 7] &

A341227 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^8.

Original entry on oeis.org

1, 16, 136, 824, 4004, 16608, 61076, 204200, 631714, 1831752, 5027312, 13159104, 33049090, 80030808, 187613348, 427201176, 947520103, 2051989360, 4347996772, 9030416704, 18412343832, 36905322248, 72807201940, 141525042736, 271321432489, 513454659312

Offset: 8

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Column k=8 of A060642.

Cf. A000041, A023007, A048574, A327386, A341221, A341222, A341223, A341225, A341226, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..33);  # Alois P. Heinz, Feb 07 2021

nmax = 33; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &

A341228 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^9.

Original entry on oeis.org

1, 18, 171, 1149, 6147, 27891, 111567, 403722, 1345896, 4189334, 12300174, 34337403, 91721385, 235645425, 584759880, 1406588073, 3289489002, 7498465029, 16697615817, 36391839264, 77758115283, 163123713621, 336420277812, 682877289213, 1365674365197, 2693384989056

Offset: 9

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..10000

Column k=9 of A060642.

Cf. A000041, A023008, A048574, A327387, A341221, A341222, A341223, A341225, A341226, A341227.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..34);  # Alois P. Heinz, Feb 07 2021

nmax = 34; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &

A341365 Expansion of (1 / theta_4(x) - 1)^4 / 16.

Original entry on oeis.org

1, 8, 40, 156, 520, 1552, 4262, 10960, 26716, 62276, 139744, 303412, 640001, 1315832, 2644004, 5204044, 10052182, 19086348, 35672516, 65708116, 119409576, 214289116, 380068582, 666723748, 1157550524, 1990230968, 3390558072, 5726064688, 9590759624, 15938198484, 26289242026

Offset: 4

Ilya Gutkovskiy, Feb 10 2021

nonn

Robert Israel, Table of n, a(n) for n = 4..10000

Cf. A002448, A004405, A014968, A015128, A063730, A284286, A327382, A338223, A341222, A341364, A341366, A341367, A341368, A341369, A341370.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..34);  # Alois P. Heinz, Feb 10 2021

nmax = 34; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^4/16, {x, 0, nmax}], x] // Drop[#, 4] &
nmax = 34; CoefficientList[Series[(1/16) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: (1/16) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^4.

a(n) ~ A284286(n)/16. - Vaclav Kotesovec, Feb 20 2021

A341236 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^10.

Original entry on oeis.org

1, 20, 210, 1550, 9055, 44624, 192945, 751480, 2686155, 8934560, 27946335, 82884860, 234636435, 637416140, 1669127130, 4228739712, 10398140075, 24882425770, 58080468790, 132508486900, 296005537183, 648445364080, 1394961003490, 2950516502980, 6142674032345, 12599932782780

Offset: 10

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..10000

Column k=10 of A060642.

Cf. A000041, A023009, A048574, A327388, A341221, A341222, A341223, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..35);  # Alois P. Heinz, Feb 07 2021

nmax = 35; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &

cp's OEIS Frontend

A341243 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^4.

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A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n.

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A341221 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^3.

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A341223 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^5.

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A341225 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^6.

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A341226 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^7.

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A341227 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^8.

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A341228 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^9.

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A341365 Expansion of (1 / theta_4(x) - 1)^4 / 16.