M.S. Mahadeva Naika - cp's OEIS frontend

User: M.S. Mahadeva Naika

M.S. Mahadeva Naika has authored 3 sequences.

A273228 G.f. is the fourth power of the g.f. of A006950.

1, 4, 10, 24, 55, 116, 230, 440, 819, 1480, 2602, 4480, 7580, 12604, 20620, 33272, 53029, 83520, 130088, 200600, 306488, 464168, 697150, 1039032, 1537435, 2259300, 3298428, 4785880, 6903657, 9903040, 14129846, 20058488, 28336790, 39845456, 55778050, 77747328, 107924347, 149221160

Offset: 0

M.S. Mahadeva Naika, May 18 2016

nonn

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
L. Wang, Arithmetic properties of partition triples with odd parts distinct, Int. J. Number Theory, 11 (2015), 1791--1805.
L. Wang, Arithmetic properties of partition quadruples with odd parts distinct, Bull. Aust. Math. Soc., doi:10.1017/S0004972715000647.
L. Wang, New congruences for partitions where the odd parts are distinct, J. Integer Seq. (2015), article 15.4.2.

Cf. A006960, A273226.

Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
GenFun:=series(etaq(q,2,1000)^4/etaq(q,1,1000)^4/etaq(q,4,1000)^4,q,50):
CoefficientList(sort(convert(GenFun,polynom),q,ascending),q);

nmax = 30; CoefficientList[Series[Product[(1 + x^k)^4 / (1 - x^(4*k))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 25 2017 *)
CoefficientList[Series[1/(QPochhammer[q, -q]*QPochhammer[q^2, q^2])^4, {q, 0, 50}], q] (* G. C. Greubel, Apr 17 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^4 / (1 - x^(4*k))^4, corrected by Vaclav Kotesovec, Mar 25 2017
Expansion of 1 / psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.
a(n) ~ exp(sqrt(2*n)*Pi) / (2^(9/4)*n^(7/4)). - Vaclav Kotesovec, Mar 25 2017

Edited by N. J. A. Sloane, May 26 2016

A273226 G.f. is the cube of the g.f. of A006950.

Original entry on oeis.org

1, 3, 6, 13, 27, 51, 91, 159, 273, 455, 738, 1179, 1860, 2886, 4410, 6667, 9981, 14781, 21671, 31512, 45474, 65113, 92547, 130689, 183439, 255930, 355017, 489895, 672672, 919152, 1250107, 1692846, 2282895, 3066180, 4102224, 5468160, 7263217, 9614436, 12684633, 16682276

Offset: 0

M.S. Mahadeva Naika, May 18 2016

nonn

Robert Israel, Table of n, a(n) for n = 0..10000
M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
L. Wang, Arithmetic properties of partition triples with odd parts distinct, Int. J. Number Theory, 11 (2015), 1791--1805.
L. Wang, Arithmetic properties of partition quadruples with odd parts distinct, Bull. Aust. Math. Soc., doi:10.1017/S0004972715000647.
L. Wang, New congruences for partitions where the odd parts are distinct, J. Integer Seq. (2015), article 15.4.2.

Cf. A006950.

N:= 50:
G:= mul((1+x^k)^3,k=1..N)/mul((1-x^(4*k))^3,k=1..N/4):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 21 2019

s = QPochhammer[-1, x]^3/(8*QPochhammer[x^4, x^4]^3) + O[x]^40; CoefficientList[s, x] (* Jean-François Alcover, May 20 2016 *)

G.f.: Product_{k>=1} (1 + x^k)^3 / (1 - x^(4*k))^3, corrected by Vaclav Kotesovec, Mar 25 2017.

a(n) ~ 3*exp(sqrt(3*n/2)*Pi) / (16*n^(3/2)). - Vaclav Kotesovec, Mar 25 2017

Edited by N. J. A. Sloane, May 26 2016

A273225 Number of bipartitions of n wherein odd parts are distinct (and even parts are unrestricted).

Original entry on oeis.org

1, 2, 3, 6, 11, 18, 28, 44, 69, 104, 152, 222, 323, 460, 645, 902, 1254, 1722, 2343, 3174, 4278, 5722, 7601, 10056, 13250, 17358, 22623, 29382, 38021, 48984, 62857, 80404, 102528, 130282, 165002, 208398, 262495, 329666, 412878, 515840

Offset: 0

M.S. Mahadeva Naika, May 18 2016

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of bipartitions of 'n' wherein odd parts are distinct (and even parts are unrestricted).
G.f. is the square of the g.f. of A006950. - Vaclav Kotesovec, Mar 25 2017

			a(4)=11 because "(0,4)=(0,3+1)=(0,2+2)=(1,3)=(1,2+1)=(2,2)=(4,0)=(3+1,0)=(2+2,0)=(3,1)=(2+1,1)".
G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 28*x^6 + 44*x^7 + ... - _Michael Somos_, Mar 02 2019
G.f. = q^-1 + 2*q^3 + 3*q^7 + 6*q^11 + 11*q^15 + 18*q^19 + 28*q^23 + ... - _Michael Somos_, Mar 02 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
Michael Somos, Introduction to Ramanujan theta functions
L. Wang, Arithmetic properties of partition triples with odd parts distinct, Int. J. Number Theory, 11 (2015), 1791--1805.
L. Wang, Arithmetic properties of partition quadruples with odd parts distinct, Bull. Aust. Math. Soc., doi:10.1017/S0004972715000647.
L. Wang, New congruences for partitions where the odd parts are distinct, J. Integer Seq. (2015), article 15.4.2.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

For a version with signs see A274621.

Cf. A006950.

Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
GenFun:=series(etaq(q,2,100)^2/etaq(q,1,100)^2/etaq(q,4,100)^2,q,50):
CoefficientList(sort(convert(GenFun,polynom),q,ascending),q);

s = QPochhammer[-1, x]^2/(4*QPochhammer[x^4, x^4]^2) + O[x]^40; CoefficientList[s, x] (* Jean-François Alcover, May 20 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2, x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, Mar 02 2019 *)

{a(n) = my(A); if( n<0, 0 , A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A))^2, n))}; /* Michael Somos, Mar 02 2019 */

G.f.: Product_{k>=1} (1 + x^k)^2 / (1 - x^(4*k))^2, corrected by Vaclav Kotesovec, Mar 25 2017
Expansion of 1 / psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.
a(n) ~ exp(Pi*sqrt(n))/(2^(5/2)*n^(5/4)). - Vaclav Kotesovec, Jul 05 2016
Euler transform of period 4 sequence [2, 0, 2, 2, ...]. - Michael Somos, Mar 02 2019

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User: M.S. Mahadeva Naika

A273228 G.f. is the fourth power of the g.f. of A006950.

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A273226 G.f. is the cube of the g.f. of A006950.

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A273225 Number of bipartitions of n wherein odd parts are distinct (and even parts are unrestricted).

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