A170956 -id:A170956 - cp's OEIS frontend

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

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

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 *)

A170960 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 7.

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

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..105 (full sequence)

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

a(n) = a(105-n). - Rick L. Shepherd, Mar 01 2013

A170961 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 8.

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

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..136 (full sequence)

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

CoefficientList[Series[Product[1+x^(4i-1),{i,8}],{x,0,110}],x] (* Harvey P. Dale, Aug 22 2012 *)

a(n) = a(136-n). - Rick L. Shepherd, Mar 01 2013

A170962 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 9.

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

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..171 (full sequence)

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

a(n) = a(171-n). - Rick L. Shepherd, Mar 01 2013

A170963 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 10.

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, 0, 2, 7, 4, 0, 3, 8, 4, 0, 5, 9, 3, 1, 6, 10, 3, 1, 9, 10, 2, 2, 10, 10, 2, 3, 13, 10, 1, 5, 14, 9, 1, 7, 16, 8, 1, 9, 16, 7, 1, 11, 18, 5, 2, 14, 16, 4, 3, 16, 16, 3, 5, 18, 14

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..210 (full sequence)

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

a(n) = a(210-n). - Rick L. Shepherd, Mar 01 2013

A170964 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 11.

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, 0, 3, 8, 5, 0, 5, 10, 4, 1, 6, 11, 4, 1, 9, 12, 3, 2, 11, 12, 3, 3, 14, 13, 2, 5, 16, 12, 2, 7, 19, 12, 2, 10, 20, 11, 2, 12, 23, 10, 2, 16, 23, 8, 3, 19, 24, 7, 5

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..253 (full sequence)

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

CoefficientList[Series[Product[1+x^(4k-1),{k,11}],{x,0,100}],x] (* Harvey P. Dale, Sep 19 2020 *)

a(n) = a(253-n). - Rick L. Shepherd, Mar 01 2013

A170957 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 4.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 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.

A170958 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 5.

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, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 2, 1, 0, 1, 2, 0, 0, 1, 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.

CoefficientList[Series[Product[1+x^(4i-1),{i,5}],{x,0,60}],x] (* Harvey P. Dale, Mar 09 2019 *)

A170959 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 6.

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, 0, 0, 2, 2, 0, 0, 3, 2, 0, 1, 3, 1, 0, 1, 3, 1, 0, 2, 3, 0, 0, 2, 2, 0, 0, 3, 1, 0, 1, 2, 1, 0, 1, 2, 0, 0, 1, 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.

cp's OEIS Frontend

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

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

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A170960 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 7.

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A170961 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 8.

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A170962 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 9.

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A170963 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 10.

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A170964 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 11.

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A170957 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 4.

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A170958 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 5.

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