A001935 -id:A001935 - cp's OEIS frontend

A296068 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j))/(1 - x^(2j-1)))^k.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 4, 0, 1, 5, 14, 22, 18, 6, 0, 1, 6, 20, 40, 48, 32, 9, 0, 1, 7, 27, 65, 101, 99, 55, 12, 0, 1, 8, 35, 98, 185, 236, 194, 90, 16, 0, 1, 9, 44, 140, 309, 481, 518, 363, 144, 22, 0, 1, 10, 54, 192, 483, 882, 1165, 1080, 657, 226, 29, 0, 1, 11, 65, 255, 718, 1498, 2330, 2665, 2162, 1155, 346, 38, 0

Offset: 0

Ilya Gutkovskiy, Dec 04 2017

			G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 3)*x^2 + (1/6)*k*(k^2 + 9*k + 8)*x^3 + (1/24)*k*(k^3 + 18*k^2 + 59*k + 18)*x^4 + (1/120)*k*(k^4 + 30*k^3 + 215*k^2 + 330*k + 144)*x^5 + ...
Square array begins:
1,  1,   1,   1,    1,    1,  ...
0,  1,   2,   3,    4,    5,  ...
0,  2,   5,   9,   14,   20,  ...
0,  3,  10,  22,   40,   65,  ...
0,  4,  18,  48,  101,  185,  ...
0,  6,  32,  99,  236,  481,  ...

Columns k=0..8 give A000007, A001935, A001936, A001937, A093160, A001939, A001940, A001941, A092877.
Main diagonal gives A296044.
Antidiagonal sums give A302020.
Cf. A296067.

Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i))/(1 - x^(2 i - 1)))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[((1 - x^(4 i))/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[2^(-k/2) (EllipticTheta[2, 0, x]/(x^(1/8) EllipticTheta[2, Pi/4, Sqrt[x]]))^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^(2*j))/(1 - x^(2*j-1)))^k.
G.f. of column k: Product_{j>=1} ((1 - x^(4*j))/(1 - x^j))^k.
G.f. of column k: 2^(-k/2)*(theta_2(0,x)/(x^(1/8)*theta_2(Pi/4,sqrt(x))))^k, where theta_() is the Jacobi theta function.

A339406 Number of partitions of n into an even number of parts that are not multiples of 4.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 5, 5, 10, 9, 16, 17, 29, 28, 44, 48, 73, 76, 110, 121, 172, 185, 253, 282, 381, 417, 549, 616, 802, 889, 1137, 1279, 1620, 1810, 2260, 2549, 3161, 3544, 4346, 4906, 5979, 6720, 8120, 9164, 11014, 12392, 14788, 16682, 19820, 22297, 26337, 29682, 34921, 39267

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(6) = 5 because we have [5, 1], [3, 3], [3, 1, 1, 1], [2, 2, 1, 1] and [1, 1, 1, 1, 1, 1].

Cristina Ballantine and Mircea Merca, 4-Regular partitions and the pod function, arXiv:2111.10702 [math.CO], 2021.
Index entries for sequences related to partitions

Cf. A001935, A027187, A042968, A261734, A339404, A339405, A339407.

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
      b(n, i-1, t)+`if`(irem(i, 4)=0, 0, b(n-i, min(n-i, i), 1-t))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 03 2020

nmax = 53; CoefficientList[Series[(1/2) (Product[(1 - x^(4 k))/(1 - x^k), {k, 1, nmax}] + Product[(1 + x^(4 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(n) = (A001935(n) + A261734(n)) / 2.

A339407 Number of partitions of n into an odd number of parts that are not multiples of 4.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 4, 7, 6, 13, 13, 21, 21, 36, 38, 57, 59, 90, 98, 137, 148, 210, 231, 310, 341, 459, 511, 664, 737, 957, 1073, 1357, 1518, 1918, 2156, 2673, 3002, 3712, 4182, 5100, 5737, 6976, 7866, 9460, 10652, 12777, 14402, 17126, 19284, 22867, 25761, 30340, 34139, 40099

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(6) = 4 because we have [6], [3, 2, 1], [2, 2, 2] and [2, 1, 1, 1, 1].

Cristina Ballantine and Mircea Merca, 4-Regular partitions and the pod function, arXiv:2111.10702 [math.CO], 2021.
Index entries for sequences related to partitions

Cf. A001935, A027193, A042968, A261734, A339404, A339405, A339406.

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
      b(n, i-1, t)+`if`(irem(i, 4)=0, 0, b(n-i, min(n-i, i), 1-t))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 03 2020

nmax = 53; CoefficientList[Series[(1/2) (Product[(1 - x^(4 k))/(1 - x^k), {k, 1, nmax}] - Product[(1 + x^(4 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(n) = (A001935(n) - A261734(n)) / 2.

A376853 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

Original entry on oeis.org

1, 1, 4, 9, 16, 28, 49, 84, 140, 228, 361, 560, 856, 1288, 1916, 2821, 4108, 5928, 8480, 12024, 16920, 23637, 32788, 45196, 61928, 84368, 114332, 154160, 206857, 276308, 367476, 486680, 641996, 843656, 1104592, 1441168, 1873965, 2428816, 3138132, 4042408, 5192132

Offset: 0

Vaclav Kotesovec, Oct 06 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001935, A143184, A192918, A376812, A376852, A376854.

nmax = 60; CoefficientList[Series[Sum[x^(k*(k+1)/2) * Product[(1+x^j)/(1-x^j), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

a(n) ~ c * exp(sqrt(8*n*(log(r)^2 + polylog(2,r) - polylog(2,-r)))), where r = A192918 = 0.54368901269207636157... is the real root of the equation r*(1+r^2) = (1-r^2) and c = 0.0643033662740307713580663125340126524175...

A117274 Triangle read by rows: T(n,k) is the number of partitions of n with no even part repeated and having k 1's (n>=0, 0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 3, 2, 1, 1, 1, 0, 1, 3, 3, 2, 1, 1, 1, 0, 1, 4, 3, 3, 2, 1, 1, 1, 0, 1, 6, 4, 3, 3, 2, 1, 1, 1, 0, 1, 7, 6, 4, 3, 3, 2, 1, 1, 1, 0, 1, 9, 7, 6, 4, 3, 3, 2, 1, 1, 1, 0, 1, 12, 9, 7, 6, 4, 3, 3, 2, 1, 1, 1, 0, 1, 14, 12, 9, 7, 6, 4, 3, 3, 2, 1, 1, 1, 0

Offset: 0

Emeric Deutsch, Mar 06 2006

Row sums yield A001935. T(n,0)=A117275(n). T(n,k)=T(n-k,0)=A117275(n-k). Sum(k*T(n,k),k=0..n)=A117276(n).

			T(8,2)=3 because we have [6,1,1],[4,2,1,1] and [3,3,1,1].

Cf. A001935, A117275, A117276.

g:=(1+x^2)*product((1+x^(2*k))/(1-x^(2*k-1)),k=2..50)/(1-t*x): gser:=simplify(series(g,x=0,23)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

G.f.=G(t,x)=(1+x^2)*product((1+x^(2k))/(1-x^(2k-1)), k=2..infinity)/(1-tx).

A117276 Number of 1's in all partitions of n with no even parts repeated.

Original entry on oeis.org

0, 1, 2, 4, 7, 11, 17, 26, 38, 54, 76, 105, 143, 193, 257, 339, 444, 576, 742, 950, 1208, 1528, 1923, 2407, 2999, 3721, 4597, 5657, 6937, 8476, 10322, 12532, 15168, 18306, 22034, 26450, 31672, 37835, 45091, 53619, 63625, 75341, 89037, 105023, 123647

Offset: 0

Emeric Deutsch, Mar 06 2006

nonn

a(n)=Sum(k*A117274(n,k),k=0..n).

			a(5)=11 because the partitions of 5 with no even parts repeated are [5],[4,1],[3,2],[3,1,1],[2,1,1,1] and [1,1,1,1,1] and they have a total number 0+1+0+2+3+5=11 parts equal to 1.

Cf. A001935, A117274.

g:=x*product((1+x^(2*j))/(1-x^(2*j-1)),j=1..35)/(1-x): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..47);

nmax = 50; CoefficientList[Series[x/(1-x) * Product[(1+x^(2*k))/(1-x^(2*k-1)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)

G.f.: x*product((1+x^(2j))/(1-x^(2j-1)), j=1..infinity)/(1-x).
a(n) ~ exp(sqrt(n/2)*Pi) / (2^(5/4)*Pi*n^(1/4)). - Vaclav Kotesovec, Mar 07 2016
G.f.: (x/(1 - x))*Product_{k>=1} (1 - x^(4*k))/(1 - x^k). - Ilya Gutkovskiy, May 15 2018

A226034 Expansion of f(-x)^6 / (chi(x) * phi(-x)^6) in powers of x where phi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 11, 73, 368, 1552, 5755, 19337, 60054, 174801, 481760, 1266992, 3198963, 7791921, 18382187, 42139440, 94126547, 205343040, 438390320, 917501570, 1885269635, 3808353889, 7571955531, 14833349529, 28657374307, 54646711136, 102932171227, 191644299945

Offset: 0

Michael Somos, May 28 2013

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 + 11*x + 73*x^2 + 368*x^3 + 1552*x^4 + 5755*x^5 + 19337*x^6 + 60054*x^7 + ...
q^19 + 11*q^43 + 73*q^67 + 368*q^91 + 1552*q^115 + 5755*q^139 + 19337*q^163 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001935.

nmax=60; CoefficientList[Series[Product[(1+x^k)^4 * (1-x^(3*k))^6 * (1-x^(4*k)) / (1-x^k)^7,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-19/24)* eta[q^2]^4*eta[q^3]^6*eta[q^4]/eta[q]^11, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 15 2018 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^6 * eta(x^4 + A) / eta(x + A)^11, n))}

Expansion of q^(-19/24) * eta(q^2)^4 * eta(q^3)^6 * eta(q^4) / eta(q)^11 in powers of q.
a(n) = 1/12 * A001935(9*n + 7).
a(n) ~ exp(3*Pi*sqrt(n/2)) / (2^(19/4) * 3^(5/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

A293195 E.g.f.: Product_{m>0} (1+x^m+x^(2m)/2!+x^(3m)/3!).

Original entry on oeis.org

1, 1, 3, 13, 72, 500, 4020, 37380, 389760, 4546080, 58363200, 814968000, 12301027200, 200216016000, 3484710028800, 64639070496000, 1270187702784000, 26414731639296000, 578733086131200000, 13328586071184384000, 321801976039864320000, 8127599117746268160000

Offset: 0

Seiichi Manyama, Oct 02 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=3 of A293135.

Cf. A001935.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)/j!, j=0..min(3, n/i))))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 02 2017

A332310 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 4.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 9, 5, 12, 17, 23, 43, 50, 55, 67, 111, 144, 273, 291, 377, 410, 689, 827, 961, 1860, 1663, 2647, 3573, 4610, 4683, 6753, 8465, 11232, 16835, 19985, 24073, 29258, 40411, 51367, 58431, 72084, 99807, 119409, 176387, 199922, 250841, 290123

Offset: 0

Ilya Gutkovskiy, Feb 09 2020

nonn

			a(7) = 5 because we have [7], [6, 1], [5, 2], [2, 5] and [1, 6].

Index entries for sequences related to compositions

Cf. A001935, A032020, A032021, A070048, A332309, A332311.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 09 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!, Sum[b[n - i j, i - 1, p + j], {j, 0, Min[Mod[i, 4], 1, n/i]}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 55] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

A344319 Number of partitions of n into consecutive parts not divisible by 4.

Original entry on oeis.org

1, 1, 2, 0, 2, 2, 1, 1, 1, 2, 3, 0, 2, 2, 1, 2, 2, 2, 2, 0, 3, 2, 2, 2, 1, 2, 3, 0, 2, 3, 1, 3, 2, 2, 2, 0, 4, 2, 1, 2, 1, 3, 5, 0, 2, 2, 1, 3, 1, 3, 3, 0, 4, 3, 1, 2, 2, 2, 3, 0, 3, 2, 2, 4, 1, 3, 3, 0, 3, 4, 1, 2, 1, 2, 4, 0, 3, 3, 2, 4, 2, 2, 3, 0, 4, 2, 1, 2, 2, 4

Offset: 1

Ilya Gutkovskiy, May 14 2021

nonn

			a(11) = 3 because we have [11], [6, 5] and [5, 3, 2, 1].

Cf. A001227, A001935, A034178, A042968, A344318, A344320.

cp's OEIS Frontend

A296068 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2*j))/(1 - x^(2*j-1)))^k.

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A339406 Number of partitions of n into an even number of parts that are not multiples of 4.

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Maple

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A339407 Number of partitions of n into an odd number of parts that are not multiples of 4.

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A376853 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

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A117274 Triangle read by rows: T(n,k) is the number of partitions of n with no even part repeated and having k 1's (n>=0, 0<=k<=n).

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Maple

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A117276 Number of 1's in all partitions of n with no even parts repeated.

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Maple

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A226034 Expansion of f(-x)^6 / (chi(x) * phi(-x)^6) in powers of x where phi(), chi(), f() are Ramanujan theta functions.

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A293195 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!+x^(3*m)/3!).

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A296068 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j))/(1 - x^(2j-1)))^k.

A376853 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

A293195 E.g.f.: Product_{m>0} (1+x^m+x^(2m)/2!+x^(3m)/3!).