A053261 -id:A053261 - cp's OEIS frontend

A053257 Coefficients of the '5th-order' mock theta function f_1(q).

1, 0, 1, -1, 1, -1, 2, -2, 1, -1, 2, -2, 2, -2, 2, -3, 3, -2, 3, -4, 4, -4, 4, -5, 5, -4, 5, -6, 6, -6, 7, -8, 7, -7, 8, -9, 10, -9, 10, -12, 11, -11, 13, -14, 14, -15, 16, -17, 17, -16, 19, -21, 20, -21, 23, -25, 25, -25, 27, -29, 30, -30, 32, -35, 35, -35, 39, -41, 41, -43, 45, -49, 50, -49, 53, -57, 58, -59, 63, -67, 68

Offset: 0

Dean Hickerson, Dec 19 1999

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Other '5th-order' mock theta functions are at A053256, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

Series[Sum[q^(n^2+n)/Product[1+q^k, {k, 1, n}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k^2+k) / Product[1+x^j, {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)

G.f.: f_1(q) = Sum_{n>=0} q^(n^2+n)/((1+q)(1+q^2)...(1+q^n)).
Consider partitions of n into parts differing by at least 2 and with smallest part at least 2. a(n) is the number of them with largest part even minus number with largest part odd.
a(n) ~ (-1)^n * sqrt(phi) * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 15 2019

A053259 Coefficients of the '5th-order' mock theta function phi_1(q).

Original entry on oeis.org

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

Offset: 0

Dean Hickerson, Dec 19 1999

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22, 25.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

Series[Sum[q^(n+1)^2 Product[1+q^(2k-1), {k, 1, n}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^((k+1)^2) * Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

G.f.: phi_1(q) = Sum_{n>=0} q^(n+1)^2 (1+q)(1+q^3)...(1+q^(2n-1)).
a(n) is the number of partitions of n into odd parts such that each occurs at most twice, the largest part is unique and if k occurs as a part then all smaller positive odd numbers occur.
a(n) ~ exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A333179 G.f.: Sum_{k>=0} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 10 2020

nonn

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

Cf. A003106, A053261, A306734.

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[1+x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

a(n) ~ c * A333198^sqrt(n) / sqrt(n), where c = 0.3207396095989103757477946185... = sqrt((1 - (2/(23*(23 + 3*sqrt(69))))^(1/3) + ((1/2)*(23 + 3*sqrt(69)))^(1/3)/23^(2/3))/3)/2, c = sqrt(s)/2, where s is the real root of the equation -1 + 8*s - 23*s^2 + 23*s^3 = 0.

Limit_{n->infinity} A306734(n) / a(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...

A333374 G.f.: Sum_{k>=1} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)/(1 - x^j)).

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 3, 4, 6, 8, 10, 12, 15, 18, 22, 28, 34, 42, 52, 62, 75, 90, 106, 126, 150, 176, 208, 246, 288, 338, 397, 462, 538, 626, 724, 838, 968, 1114, 1282, 1474, 1690, 1936, 2217, 2532, 2890, 3296, 3750, 4264, 4844, 5492, 6222, 7042, 7958, 8986, 10138

Offset: 0

Vaclav Kotesovec, Mar 17 2020

nonn

The g.f. Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j) = Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j + 2*x^j) == Sum_{k >= 1} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

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

Cf. A001935, A001936, A003106, A053261, A066447, A003114, A306734, A333179.

Cf. A192918, A376841.

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[(1+x^k)/(1-x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Limit_{n->infinity} A066447(n) / a(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... (the tribonacci constant).
Compare with: A306734(n) / A333179(n) -> A060006 (the plastic constant) and A003114(n) / A003106(n) -> A001622 (golden ratio).
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.10511708841962944170826735560432... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 - sinh(arcsinh(sqrt(11)/4)/3) / (12*sqrt(11))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 17 2020, updated Oct 10 2024

A376631 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j)).

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 2, 0, 3, 0, 2, 1, 3, 1, 3, 1, 2, 3, 2, 3, 2, 4, 1, 5, 2, 5, 2, 6, 1, 7, 2, 7, 3, 6, 4, 7, 5, 6, 7, 6, 7, 7, 9, 5, 11, 5, 12, 6, 14, 5, 15, 6, 16, 7, 17, 7, 18, 9, 18, 11, 19, 12, 20, 14, 19, 17, 19, 19, 20, 23, 18, 27, 18, 29, 20, 32, 19

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

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

Cf. A000009, A053258, A053261, A264905, A333179, A376580, A376632, A376660.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j)).
a(n) ~ c * A376660^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.39098976711379944962936707496887239986756106886318...
a(n) ~ A376580(n) * (A376660/A376621)^sqrt(n).

A376812 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^j)^2.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 5, 5, 5, 7, 8, 10, 13, 14, 16, 19, 21, 25, 29, 33, 40, 45, 50, 57, 64, 72, 81, 93, 104, 117, 134, 148, 165, 185, 204, 227, 253, 280, 310, 345, 381, 422, 469, 514, 567, 625, 685, 753, 825, 903, 990, 1086, 1186, 1297, 1419, 1548, 1692, 1845, 2007

Offset: 0

Vaclav Kotesovec, Oct 05 2024

nonn

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

Cf. A053261, A376626, A376627, A376813, A216222.

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1)/2)*Product[1+x^k, {k, 1, n}]^2, {n, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^k)*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

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

a(n) ~ c * A376815^sqrt(n) / sqrt(n), where c = 1/(4*sqrt(3/2 - 2*sinh(arcsinh(3^(3/2)/2)/3)/sqrt(3))) = 0.27647151570071656262813536...

A376630 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j-1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 2, 2, 0, 1, 2, 2, 1, 1, 2, 3, 2, 1, 3, 2, 2, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 5, 5, 5, 4, 4, 7, 7, 5, 6, 8, 7, 7, 6, 8, 10, 8, 8, 10, 11, 9, 10, 12, 12, 11, 12, 14, 14, 13, 13, 16, 17, 15, 17, 18, 18, 19, 19, 20, 21, 22, 22, 24, 24, 25, 26, 27, 28, 29, 30

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

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

Cf. A000009, A053258, A053261, A333179, A376629, A376660.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k - 1))*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j-1)).

a(n) ~ c * A376660^sqrt(n) / sqrt(n), where c = sqrt(cosh(arccosh(sqrt(31)/2) / 3))/31^(1/4) = 0.456748282933947534736955792823221857...

A376944 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)/2) Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 12, 8, 8, 16, 24, 24, 24, 32, 32, 64, 64, 64, 80, 80, 112, 160, 160, 160, 224, 224, 256, 320, 416, 416, 480, 576, 576, 704, 768, 896, 1152, 1216, 1280, 1536, 1600, 1856, 2112, 2304, 2560, 3200, 3456, 3584, 4224, 4480, 5120, 5760, 6144, 6656, 7808, 9088

Offset: 0

Vaclav Kotesovec, Oct 10 2024

nonn

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

Cf. A032302, A053261, A376943, A376945.

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

a(n) ~ sqrt(1 + sqrt(3)) * exp(sqrt((2*log(2)^2 + 2*log(1 - sqrt(3)/2) * log(sqrt(3) - 1) + 4*polylog(2, sqrt(3) - 1) - Pi^2/3)*n)) / (4*3^(1/4)*sqrt(n)).

A306732 Expansion of Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + j*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 2, 5, 4, 7, 8, 5, 7, 11, 15, 22, 17, 31, 39, 20, 31, 39, 64, 81, 85, 125, 97, 170, 211, 121, 167, 229, 265, 385, 531, 548, 573, 814, 686, 1150, 1339, 860, 1131, 1344, 1888, 2109, 2780, 3656, 4127, 4294, 4498, 6320, 5568, 8747, 10260, 6856, 8673, 10580

Offset: 0

Ilya Gutkovskiy, Mar 06 2019

nonn

Cf. A053261, A204856, A306704.

nmax = 59; CoefficientList[Series[Sum[x^(k (k + 1)/2) Product[(1 + j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306733 Expansion of Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^j)^j.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 3, 5, 5, 8, 11, 11, 17, 19, 22, 33, 38, 50, 69, 82, 103, 133, 165, 201, 249, 319, 389, 492, 621, 765, 974, 1206, 1500, 1857, 2302, 2843, 3494, 4311, 5275, 6533, 8027, 9840, 12138, 14903, 18340, 22541, 27619, 33811, 41429, 50682, 61809, 75422, 91807

Offset: 0

Ilya Gutkovskiy, Mar 06 2019

nonn

Cf. A053261, A206138, A306706.

nmax = 54; CoefficientList[Series[Sum[x^(k (k + 1)/2) Product[(1 + x^j)^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

A053257 Coefficients of the '5th-order' mock theta function f_1(q).

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A053259 Coefficients of the '5th-order' mock theta function phi_1(q).

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A333179 G.f.: Sum_{k>=0} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)).

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A333374 G.f.: Sum_{k>=1} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)/(1 - x^j)).

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A376631 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j)).

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A376812 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^j)^2.

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A376630 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j-1)).

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A376944 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^j).

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A306732 Expansion of Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + j*x^j).

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A333179 G.f.: Sum_{k>=0} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)).

A333374 G.f.: Sum_{k>=1} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)/(1 - x^j)).

A376631 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j)).

A376812 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^j)^2.

A376630 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j-1)).

A376944 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)/2) Product_{j=1..k} (1 + x^j).

A306732 Expansion of Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + j*x^j).

A306733 Expansion of Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^j)^j.