A169975 -id:A169975 - cp's OEIS frontend

A261612 Expansion of Product_{k>=0} (1 + x^(3*k+1)).

1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 3, 3, 2, 4, 4, 2, 4, 5, 3, 5, 7, 4, 5, 8, 6, 7, 10, 7, 7, 12, 10, 9, 14, 12, 10, 16, 16, 13, 19, 19, 15, 22, 24, 19, 25, 28, 22, 29, 35, 28, 33, 40, 33, 38, 48, 41, 44, 55, 48, 51, 66, 59, 58, 74, 69

Offset: 0

Vaclav Kotesovec, Aug 26 2015

nonn

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

Cf. A000009, A000700, A035382, A169975.

Cf. A015128, A080054, A261610, A261611, A262928.

nmax = 100; CoefficientList[Series[Product[(1 + x^(3*k+1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 3] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 13 2017 *)

a(n) ~ exp(Pi*sqrt(n)/3) / (2^(4/3) * sqrt(3) * n^(3/4)) * (1 - (Pi/144 + 9/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 26 2015, extended Jan 16 2017

G.f.: Sum_{k>=0} x^(k*(3*k - 1)/2) / Product_{j=1..k} (1 - x^(3*j)). - Ilya Gutkovskiy, Nov 24 2020

A147599 Expansion of Product_{i>=1} (1+x^(4*i-1)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 2, 3, 1, 0, 3, 4, 1, 1, 4, 4, 1, 1, 5, 5, 1, 2, 7, 5, 1, 3, 8, 6, 1, 5, 10, 6, 2, 6, 12, 7, 2, 9, 14, 7, 3, 11, 16, 8, 4, 15, 19, 8, 6, 18, 21, 9, 8, 23, 24, 10, 11, 27, 27, 11, 14, 34, 30, 12, 19, 39, 33, 14, 24, 47

Offset: 0

N. J. A. Sloane, Aug 29 2010

nonn

Number of partitions into distinct parts 4*k+3.

Convolution of A147599 and A169975 is A000700. - Vaclav Kotesovec, Jan 18 2017

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

Cf. A000700, A169975, A170956-A170965.

Cf. A262928, A281243, A281244, A281245.

nmax = 200; CoefficientList[Series[Product[(1 + x^(4*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 18 2017 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 4] == 3, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f. sum(n>=0, x^(2*n^2+n) / prod(k=1,n, 1-x^(4*k))) - Joerg Arndt, Mar 10 2011.

a(n) ~ exp(sqrt(n/3)*Pi/2) / (4*6^(1/4)*n^(3/4)) * (1 - (3*sqrt(3)/(4*Pi) + Pi/(192*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017

A281243 Expansion of Product_{k>=1} (1 + x^(5*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 1, 0, 0, 2, 3, 1, 0, 0, 3, 4, 1, 0, 1, 4, 4, 1, 0, 1, 5, 5, 1, 0, 2, 7, 5, 1, 0, 3, 8, 6, 1, 0, 5, 10, 6, 1, 1, 6, 12, 7, 1, 1, 9, 14, 7, 1, 2, 11, 16, 8, 1

Offset: 0

Vaclav Kotesovec, Jan 18 2017

nonn

Convolution of this sequence and A280454 is A203776.

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

Cf. A262928, A147599, A281244, A281245.

Cf. A261612, A169975, A280454, A280456, A280457.

nmax = 200; CoefficientList[Series[Product[(1 + x^(5*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 5] == 4, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/15)*Pi) / (2^(9/5)*15^(1/4)*n^(3/4)) * (1 + (Pi/(240*sqrt(15)) - 3*sqrt(15)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

G.f.: Sum_{k>=0} x^(k*(5*k + 3)/2) / Product_{j=1..k} (1 - x^(5*j)). - Ilya Gutkovskiy, Nov 24 2020

A280454 Expansion of Product_{k>=0} (1 + x^(5*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 2, 0, 0, 1, 3, 3, 1, 0, 1, 4, 4, 1, 0, 1, 4, 5, 2, 0, 1, 5, 7, 3, 0, 1, 5, 8, 5, 1, 1, 6, 10, 6, 1, 1, 6, 12, 9, 2, 1, 7, 14, 11, 3, 1, 7, 16, 15, 5, 1, 8, 19, 18, 7, 2, 8, 21, 23, 10, 2, 9, 24, 27, 13, 3, 9, 27, 34, 18, 4, 10, 30, 39, 23, 6, 10

Offset: 0

Ilya Gutkovskiy, Jan 03 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 5.

Convolution of A281243 and A280454 is A203776. - Vaclav Kotesovec, Jan 18 2017

			a(27) = 3 because we have [26, 1], [21, 6] and [11, 16].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
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.
Index entries for related partition-counting sequences

Cf. A000700, A016861, A109697, A169975, A261612, A280456, A280457.

Cf. A262928, A147599, A281243, A281244, A281245.

nmax = 102; CoefficientList[Series[Product[(1 + x^(5 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 5] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Product_{k>=0} (1 + x^(5*k+1)).

a(n) ~ exp(Pi*sqrt(n)/sqrt(15))/(2*2^(1/5)*15^(1/4)*n^(3/4)) * (1 + (Pi/(240*sqrt(15)) - 3*sqrt(15)/(8*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 24 2017

A281245 Expansion of Product_{k>=1} (1 + x^(7*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 2, 7, 5, 1, 0, 0, 0

Offset: 0

Vaclav Kotesovec, Jan 18 2017

nonn

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

Cf. A262928, A147599, A281243, A281244.
Cf. A261612, A169975, A280454.
Cf. A109708, A281455, A281456, A281457, A281458, A280457, A281459.

nmax = 200; CoefficientList[Series[Product[(1 + x^(7*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 7] == 6, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/21)*Pi) / (2^(13/7)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(336*sqrt(21)) - 3*sqrt(21)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

G.f.: Sum_{k>=0} x^(k*(7*k + 5)/2) / Product_{j=1..k} (1 - x^(7*j)). - Ilya Gutkovskiy, Nov 24 2020

A280457 Expansion of Product_{k>=0} (1 + x^(7*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 1, 4, 5, 2, 0, 0, 0, 1, 5, 7, 3, 0, 0, 0, 1, 5, 8, 5, 1, 0, 0, 1, 6, 10, 6, 1, 0, 0, 1, 6, 12, 9, 2, 0, 0, 1, 7, 14, 11, 3, 0, 0, 1, 7, 16, 15, 5

Offset: 0

Ilya Gutkovskiy, Jan 03 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 7.

			a(37) = 3 because we have [36, 1], [29, 8] and [22, 15].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
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.
Index entries for related partition-counting sequences

Cf. A000700, A016993, A169975, A261612, A280454.
Cf. A262928, A147599, A281243, A281244.
Cf. A109703, A281245, A281455, A281456, A281457, A281458, A281459.

nmax = 105; CoefficientList[Series[Product[(1 + x^(7 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 7] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Product_{k>=0} (1 + x^(7*k+1)).

a(n) ~ exp(Pi*sqrt(n)/sqrt(21))/(2*2^(1/7)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(336*sqrt(21)) - 3*sqrt(21)/(8*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 24 2017

A170956 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 3.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1

Offset: 0

N. J. A. Sloane, Aug 29 2010

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Cf. A000700, A147599, A169975, A170956-A170965.

A170965 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 12.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 2, 3, 1, 0, 3, 4, 1, 1, 4, 4, 1, 1, 5, 5, 1, 2, 7, 5, 1, 3, 8, 6, 0, 5, 10, 5, 1, 6, 12, 5, 1, 9, 13, 4, 2, 11, 14, 4, 3, 15, 15, 3, 5, 17, 15, 3, 7, 21, 15, 3, 10, 23, 15, 3, 13, 27, 14, 3, 17, 28, 13, 4, 21, 31, 12

Offset: 0

N. J. A. Sloane, Aug 29 2010

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Nathaniel Johnston, Table of n, a(n) for n = 0..300 (full sequence)

Cf. A000700, A147599, A169975, A170956-A170965.

CoefficientList[Series[Product[1+x^(4i-1),{i,12}],{x,0,100}],x] (* Harvey P. Dale, Dec 24 2012 *)

A281244 Expansion of Product_{k>=1} (1 + x^(6*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 3, 4, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 5, 5, 1, 0, 0, 2, 7, 5, 1, 0, 0, 3, 8, 6, 1, 0, 0, 5, 10, 6, 1, 0, 1

Offset: 0

Vaclav Kotesovec, Jan 18 2017

nonn

Convolution of this sequence and A280456 is A098884.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - minor asymptotic term, (Pi/144 - 9/(4*Pi))/sqrt(2) = -0.49100125...

Cf. A262928, A147599, A281243, A281245.

Cf. A261612, A169975, A280454, A280456, A280457.

a:= proc(n) option remember; `if`(n=0, 1, add(add(
      [0$5, 1, 0$4, -1, 1][1+irem(d, 12)]*d, d=
       numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 18 2017

nmax = 200; CoefficientList[Series[Product[(1 + x^(6*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 6] == 5, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/2)*Pi/3) / (2^(25/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/144 - 9/(4*Pi)) / sqrt(2*n)).

G.f.: Sum_{k>=0} x^(k*(3*k + 2)) / Product_{j=1..k} (1 - x^(6*j)). - Ilya Gutkovskiy, Nov 24 2020

A170966 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 3.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1

Offset: 0

N. J. A. Sloane, Aug 29 2010

Cf. A169975, A170966-A170984.

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A261612 Expansion of Product_{k>=0} (1 + x^(3*k+1)).

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A147599 Expansion of Product_{i>=1} (1+x^(4*i-1)).

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A281243 Expansion of Product_{k>=1} (1 + x^(5*k-1)).

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A280454 Expansion of Product_{k>=0} (1 + x^(5*k+1)).

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A281245 Expansion of Product_{k>=1} (1 + x^(7*k-1)).

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A280457 Expansion of Product_{k>=0} (1 + x^(7*k+1)).

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A170956 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 3.

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A170965 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 12.

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A281244 Expansion of Product_{k>=1} (1 + x^(6*k-1)).

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