A213261 -id:A213261 - cp's OEIS frontend

A282919 a(n) = A000041(49*n + 47).

124754, 118114304, 24908858009, 2366022741845, 133978259344888, 5234371069753672, 154043597379576030, 3617712763867604423, 70593393646562135510, 1178875491155735802646, 17229817230617210720599, 224282898599046831034631, 2636785814481962651219075

Offset: 0

Seiichi Manyama, Feb 24 2017

nonn

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 179.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A160528, A282920, A282921, A282922, A282923, A282924, A282925, A282926, A282927, A282928, A282929, A282930, A282931, A282932.

Cf. A000041, A213261 (p(7*n + 5)), A277958, A278559 (p(25*n + 24)), this sequence (p(49*n + 47)).

Table[PartitionsP[49n+47],{n,0,12}] (* Indranil Ghosh, Feb 25 2017 *)

a(n) = numbpart(49*n+47); \\ Indranil Ghosh, Feb 25 2017

a(n) = A213261(7*n + 6) = A000041(49*n + 47).

a(n) = 2546 * 7^2 * A160528(n) + 48934 * 7^4 * A282920(n-1) + 1418989 * 7^5 * A282921(n-2) + 2488800 * 7^7 * A282922(n-3) + 2394438 * 7^9 * A282923(n-4) + 1437047 * 7^11 * A282924(n-5) + 4043313 * 7^12 * A282925(n-6) + 161744 * 7^15 * A282926(n-7) + 32136 * 7^17 * A282927(n-8) + 31734 * 7^18 * A282928(n-9) + 3120 * 7^20 * A282929(n-10) + 204 * 7^22 * A282930(n-11) + 8 * 7^24 * A282931(n-12) + 7^25 * A282932(n-13) for n >= 13.

A071746 a(n) = p(7n+5)/7 where p(k) denotes the k-th partition number.

Original entry on oeis.org

1, 11, 70, 348, 1449, 5334, 17822, 55165, 160215, 441105, 1159752, 2929465, 7142275, 16873472, 38749850, 86737678, 189672868, 405991500, 852077072, 1756048833, 3558408287, 7098041203, 13951818365, 27047831797, 51760979985

Offset: 0

Benoit Cloitre, Jun 24 2002

One of the congruences related to the partition numbers stated by Ramanujan.

Berndt and Rankin, "Ramanujan: letters and commentaries", AMS-LMS, History of mathematics, vol. 9, pp. 192-193.
G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940. - From N. J. A. Sloane, Jun 07 2012

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. L. Drost, A Shorter Proof of the Ramanujan Congruence Modulo 5, Amer. Math. Monthly 104(10) (1997), 963-964.
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6(1) (1969), 56-59. MR0236136 (38 #4434). - From _N. J. A. Sloane_, Jun 07 2012

Cf. A000041, A017041, A071734, A076394, A213261, A327582, A327714, A327770.

a:= func< n | NumberOfPartitions((7*n+5)) div 7 >; [ a(n) : n in [0..30]]; // Vincenzo Librandi, Nov 30 2015

Table[PartitionsP[7n+5]/7, {n, 0, 24}] (* Jean-François Alcover, Nov 30 2015 *)

a(n)=if(n<0, 0, n=7*n+5; polcoeff(1/eta(x+x*O(x^n)),n)/7)

{a(n)=local(A,B); if(n<0, 0, A=x*O(x^n); B=eta(x^7+A); A=eta(x+A); polcoeff( B^3/A^4 +x*7*B^7/A^8, n))} /* Michael Somos, Jan 01 2006 */

a(n) = numbpart(7*n+5)/7; \\ Michel Marcus, Nov 30 2015

a(n) = (1/7)*A000041(7n+5).

a(n) = A000041(A017041(n))/7 = A213261(n)/7. - Omar E. Pol, Jan 18 2013

A160527 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 118.

Original entry on oeis.org

1, 4, 14, 40, 105, 252, 574, 1237, 2568, 5138, 9988, 18893, 34937, 63238, 112370, 196244, 337477, 572024, 956956, 1581321, 2583637, 4176495, 6684820, 10599939, 16661401, 25972485, 40171474, 61672695, 94017765, 142368024, 214211760, 320350725, 476299978

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			G.f. = 1 + 4*x + 14*x^2 + 40*x^3 + 105*x^4 + 252*x^5 + 574*x^6 + ...
G.f. = q^17 + 4*q^41 + 14*q^65 + 40*q^89 + 105*q^113 + 252*q^137 + 574*q^161 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. A071746, A213261.

nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))^3 /(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

See Maple code in A160525 for formula.
G.f.: Product_{n >= 1} (1 - x^(7*n))^3/(1 - x^n)^4. - Seiichi Manyama, Nov 06 2016
a(n) ~ exp(Pi*sqrt(50*n/21)) * 5 / (196*sqrt(3)*n). - Vaclav Kotesovec, Nov 10 2017

A327770 a(n) = (23 * 7^(2*n) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

Original entry on oeis.org

1, 47, 2301, 112747, 5524601, 270705447, 13264566901, 649963778147, 31848225129201, 1560563031330847, 76467588535211501, 3746911838225363547, 183598680073042813801, 8996335323579097876247, 440820430855375795936101, 21600201111913414000868947

Offset: 0

Petros Hadjicostas, Sep 24 2019

If p(n) = A000041(n) is the partition function, Watson (1938) proved that p(7^(2*m)*n + a(m)) == 0 mod 7^(m+1) for n >= 0 and m >= 1. (Obviously, this is not always true for m = 0).
For m=1 and n=0, p(7^(2*1)*0 + a(1)) = p(47) = 7^(1+1) * 2546.
For m=1 and n=1, p(7^(2*1)*1 + a(1)) = p(96) = 7^(1+1) * 2410496.
For m=1 and n=2, p(7^(2*1)*2 + a(1)) = p(145) = 7^(1+1) * 508344041.
For m=2 and n=0, p(7^(2*2)*0 + a(2)) = p(2301) = 7^(2+1) * 49629361905981812695622866669844910256876089360.
Essentially the same as A052463. - R. J. Mathar, Oct 08 2019

Colin Barker, Table of n, a(n) for n = 0..500
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see pp. 118 and 124.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Wikipedia, G. N. Watson.
Index entries for linear recurrences with constant coefficients, signature (50,-49).

Cf. A052463, A071746, A213261, A327714, A327582.

CoefficientList[Series[(1 - 3 x)/((1 - x) (1 - 49 x)), {x, 0, 15}], x] (* Michael De Vlieger, Sep 27 2019 *)
LinearRecurrence[{50,-49},{1,47},20] (* Harvey P. Dale, Mar 09 2023 *)

a(n) = (23 * 7^(2*n) + 1)/24; \\ Michel Marcus, Sep 25 2019

Vec((1 - 3*x) / ((1 - x)*(1 - 49*x)) + O(x^20)) \\ Colin Barker, Sep 25 2019

From Colin Barker, Sep 25 2019: (Start)
G.f.: (1 - 3*x) / ((1 - x)*(1 - 49*x)).
a(n) = 50*a(n-1) - 49*a(n-2) for n>1.
(End)

A327582 a(n) = (17 * 7^(2*n+1) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

Original entry on oeis.org

5, 243, 11905, 583343, 28583805, 1400606443, 68629715705, 3362856069543, 164779947407605, 8074217422972643, 395636653725659505, 19386196032557315743, 949923605595308471405, 46546256674170115098843, 2280766577034335639843305, 111757562274682446352321943

Offset: 0

Petros Hadjicostas, Sep 23 2019

If p(n) = A000041(n) is the partition function, Watson (1938) proved that p(7^(2*m+1)*n + a(m)) == 0 mod 7^(m+1) for n >= 0 and m >= 1.

It is well-known that this result is true even for m = 0 (cf. A071746 and the references there).

			For m=1 and n=0, p(7^(2*1+1)*0 + a(1)) = p(243) = 133978259344888 = 7^2 * 2734250190712.
For m=1 and n=1, p(7^(2*1+1)*1 + a(1)) = p(586) = 224282898599046831034631 = 7^2 * 4577202012225445531319.

Colin Barker, Table of n, a(n) for n = 0..500
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see pp. 118 and 124.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Wikipedia, G. N. Watson.
Index entries for linear recurrences with constant coefficients, signature (50,-49).

Cf. A000041, A071746, A110375, A213261, A327714, A327770.

a(n) = (17 * 7^(2*n+1) + 1)/24; \\ Michel Marcus, Sep 25 2019

Vec((5 - 7*x) / ((1 - x)*(1 - 49*x)) + O(x^15)) \\ Colin Barker, Sep 27 2019

From Colin Barker, Sep 27 2019: (Start)
G.f.: (5 - 7*x) / ((1 - x)*(1 - 49*x)).
a(n) = 50*a(n-1) - 49*a(n-2) for n>1.
(End)

A220502 spt(7n+5) where spt(n) = A092269(n).

Original entry on oeis.org

14, 238, 1820, 10486, 48692, 196168, 706671, 2335760, 7185780, 20832168, 57385734, 151261320, 383516700, 939524019, 2231714982, 5155845968, 11614187984, 25565740130, 55096236664, 116437751108, 241655216355, 493152387294, 990688365380

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

a(n) is divisible by 7 (see A220507).

G. E. Andrews, The number of smallest parts in the partitions of n

Cf. A017041, A092269, A213261, A220485, A220503, A220507.

a(n) = A092269(A017041(n)).

A277958 Expansion of Product_{n>=1} (1 - x^(7*n))^7/(1 - x^n)^8 in powers of x.

Original entry on oeis.org

1, 8, 44, 192, 726, 2464, 7704, 22521, 62281, 164252, 415796, 1015334, 2401462, 5519640, 12363062, 27047913, 57917068, 121588588, 250638216, 507974950, 1013409244, 1992161935, 3862461694, 7392045512, 13975011909, 26116935550, 48277368020, 88320521108, 159993054081

Offset: 0

Seiichi Manyama, Nov 06 2016

nonn

			G.f.: 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 + 7704*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Wikipedia, Ramanujan's congruences

Cf. A071746, A160527, A213261.

nmax = 20; CoefficientList[Series[Product[(1 - x^(7*k))^7 /(1 - x^k)^8 , {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

G.f.: Product_{n>=1} (1 - x^(7*n))^7/(1 - x^n)^8.
A213261(n) = 7*A160527(n) + 49*a(n - 1) for n > 0 due to Ramanujan's congruences.
a(n) ~ exp(Pi*sqrt(98*n/21)) / (1372*sqrt(3)*n). - Vaclav Kotesovec, Nov 10 2017

A327771 a(n) = p(49*n + 47)/49, where p(k) denotes the k-th partition number (i.e., A000041).

Original entry on oeis.org

2546, 2410496, 508344041, 48286178405, 2734250190712, 106823899382728, 3143746885297470, 73830872731991927, 1440681502991063990, 24058683492974200054, 351628923073820626951, 4577202012225445531319, 53811955397591074514675, 577896157936323089053580

Offset: 0

Petros Hadjicostas, Sep 24 2019

nonn

Watson (1938), p. 120, proved that p(7*n + 5) == 0 (mod 7) and p(49*n + 47) == 0 (mod 49) for n >= 0, where p() = A000041(). For more general congruence results modulo a power of 7 by George Neville Watson regarding the partition function, see A327582 and A327770.

G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see p. 120.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Wikipedia, G. N. Watson.

Cf. A000041, A052462, A052463, A052465, A052466, A071746, A213261, A327714, A327582, A327770.

Table[PartitionsP[49n+47]/49,{n, 0, 13}] (* Metin Sariyar, Sep 25 2019 *)

a(n) = numbpart(49*n + 47)/49; \\ Michel Marcus, Sep 25 2019

a(n) = A000041(49*n + 47)/49.

cp's OEIS Frontend

A282919 a(n) = A000041(49*n + 47).

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A071746 a(n) = p(7n+5)/7 where p(k) denotes the k-th partition number.

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A160527 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 118.

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A327770 a(n) = (23 * 7^(2*n) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

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A327582 a(n) = (17 * 7^(2*n+1) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

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A220502 spt(7n+5) where spt(n) = A092269(n).

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A277958 Expansion of Product_{n>=1} (1 - x^(7*n))^7/(1 - x^n)^8 in powers of x.

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A327771 a(n) = p(49*n + 47)/49, where p(k) denotes the k-th partition number (i.e., A000041).