A022645
Expansion of Product_{m>=1} (1 + m*q^m)^17.
Original entry on oeis.org
1, 17, 170, 1309, 8483, 48467, 251209, 1203311, 5397330, 22890874, 92481394, 358011602, 1334253585, 4805716553, 16782510007, 56979399970, 188517704002, 609021410570, 1924506074441, 5957712195945, 18092683604856, 53965253533463, 158264095730459, 456803437466434, 1298781701177781
Offset: 0
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Coefficients(&*[(1+m*x^m)^17:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
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[seq(coeff(series(mul((1+m*q^m)^(17), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
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With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^17, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
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m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^17)) \\ G. C. Greubel, Feb 17 2018
A022646
Expansion of Product_{m>=1} (1 + m*q^m)^18.
Original entry on oeis.org
1, 18, 189, 1518, 10224, 60552, 324657, 1606050, 7429455, 32458628, 134950419, 537136776, 2056614597, 7604901990, 27248140107, 94861629852, 321652565253, 1064430256536, 3443952349385, 10911585508344, 33900910277472, 103410118026774, 310042892332701, 914572545220908
Offset: 0
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Coefficients(&*[(1+m*x^m)^18:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
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[seq(coeff(series(mul((1+m*q^m)^(18), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
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With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^18, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
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m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^18)) \\ G. C. Greubel, Feb 17 2018
A022647
Expansion of Product_{m>=1} (1 + m*q^m)^19.
Original entry on oeis.org
1, 19, 209, 1748, 12217, 74898, 414865, 2116885, 10087480, 45348041, 193814402, 792340831, 3113639744, 11808753973, 43368768307, 154674601937, 537009888061, 1818759910067, 6019901796578, 19503777943838, 61940839239196, 193067981970548, 591298084019937
Offset: 0
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Coefficients(&*[(1+m*x^m)^19:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
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[seq(coeff(series(mul((1+m*q^m)^(19), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
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With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^19, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
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m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^19)) \\ G. C. Greubel, Feb 17 2018
A022648
Expansion of Product_{m>=1} (1 + m*q^m)^20.
Original entry on oeis.org
1, 20, 230, 2000, 14485, 91804, 524710, 2758520, 13526430, 62505180, 274345784, 1150868440, 4637343915, 18022311520, 67785066390, 247453832688, 878947211030, 3044142764520, 10299271911850, 34095293204360, 110599636109572, 351997976703180, 1100401056566170
Offset: 0
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Coefficients(&*[(1+m*x^m)^20:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 17 2018
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[seq(coeff(series(mul((1+m*q^m)^(20), m=1..100),q,101),q,j),j=0..25)]; # Muniru A Asiru, Feb 18 2018
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With[{nmax = 50}, CoefficientList[Series[Product[(1 + k*q^k)^20, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Feb 17 2018 *)
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m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^20)) \\ G. C. Greubel, Feb 17 2018
A022652
Expansion of Product_{m>=1} (1+m*q^m)^24.
Original entry on oeis.org
1, 24, 324, 3248, 26802, 191904, 1230824, 7221744, 39342783, 201199888, 974039652, 4493483424, 19859122142, 84451085664, 346817307672, 1379695128080, 5330825817507, 20050294307376, 73556403409336
Offset: 0
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Coefficients(&*[(1+m*x^m)^24:m in [1..40]])[1..50] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Jul 18 2018
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With[{nmax=50}, CoefficientList[Series[Product[(1+m*q^m)^24,{m,1,nmax}],{q,0,nmax}],q]] (* G. C. Greubel, Jul 18 2018 *)
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m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^24)) \\ G. C. Greubel, Jul 18 2018
A022654
Expansion of Product_{m>=1} (1+m*q^m)^26.
Original entry on oeis.org
1, 26, 377, 4030, 35282, 267020, 1804855, 11133278, 63635364, 340845830, 1725623406, 8314033858, 38329313893, 169845329890, 726114272520, 3004404814658, 12063899757390, 47120073874016, 179388891204380, 666854279935844, 2424357631391397, 8631804737992852
Offset: 0
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Coefficients(&*[(1+m*x^m)^26:m in [1..40]])[1..50] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Jul 18 2018
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b:= proc(n) option remember; `if`(n=0, 1, add(
26*(-d)^(n/d+1), d=numtheory[divisors](n)))
end:
a:= proc(n) option remember; `if`(n=0, 1,
add(b(j)*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jul 18 2018
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With[{nmax=50}, CoefficientList[Series[Product[(1+m*q^m)^26,{m,1,nmax}],{q,0,nmax}],q]] (* G. C. Greubel, Jul 18 2018 *)
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m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^26)) \\ G. C. Greubel, Jul 18 2018
A022660
Expansion of Product_{m>=1} (1+m*q^m)^32.
Original entry on oeis.org
1, 32, 560, 7104, 72888, 640320, 4985600, 35202496, 229089692, 1390677728, 7947553824, 43070246592, 222637403008, 1103015334464, 5258564956736, 24206137227648, 107897964171910, 466891402634624, 1965526100961552, 8065514386997056, 32315388450560032
Offset: 0
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Coefficients(&*[(1+m*x^m)^32:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 24 2018
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With[{nmax=50}, CoefficientList[Series[Product[(1+k*q^k)^32, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 24 2018 *)
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m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+n*q^n)^32)) \\ G. C. Greubel, Feb 24 2018
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