A022629 -id:A022629 - cp's OEIS frontend

A303188 a(n) = [x^n] Product_{k=1..n} (1 + (n - k + 1)*x^k).

1, 1, 1, 7, 9, 23, 148, 221, 526, 1040, 6767, 9664, 23456, 43943, 91363, 499028, 736410, 1650395, 3107540, 6210372, 10819270, 57864166, 80663444, 179915133, 324882691, 640398244, 1087149284, 2039724322, 9121580902, 12913282685, 27250167385, 48645989650, 92634730208, 156124357449

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

			a(0) = 1;
a(1) = [x^1] (1 + x) = 1;
a(2) = [x^2] (1 + 2*x)*(1 + x^2) = 1;
a(3) = [x^3] (1 + 3*x)*(1 + 2*x^2)*(1 + x^3) = 7;
a(4) = [x^4] (1 + 4*x)*(1 + 3*x^2)*(1 + 2*x^3)*(1 + x^4) = 9;
a(5) = [x^5] (1 + 5*x)*(1 + 4*x^2)*(1 + 3*x^3)*(1 + 2*x^4)*(1 + x^5) = 23, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} (1 + (n - k + 1)*x^k) begins:
n = 0: (1), 0,  0,   0,   0,   0,  ...
n = 1:  1, (1), 0,   0,   0,   0,  ...
n = 2:  1,  2, (1),  2,   0,   0   ...
n = 3:  1,  3,  2,  (7),  3,   2,  ...
n = 4:  1,  4,  3,  14,  (9), 10,  ...
n = 5:  1,  5,  4,  23,  17, (23), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A022629, A206229, A291698, A303175, A303189, A303190.

Table[SeriesCoefficient[Product[(1 + (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 33}]

A318248 Expansion of Product_{k>=1} (1 + Fibonacci(k)*x^k).

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 18, 35, 63, 123, 220, 411, 750, 1387, 2498, 4649, 8308, 15150, 27446, 49638, 88754, 161280, 287831, 516770, 924956, 1655166, 2944850, 5272056, 9348047, 16631195, 29569572, 52421323, 92665614, 164437988, 290243745, 512649342, 904774082

Offset: 0

Vaclav Kotesovec, Aug 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4500
Michael Hendriksen, Nils Kapust, On the comparison of incompatibility of split systems across different taxa sizes, arXiv:2004.00062 [q-bio.PE], 2020.

Cf. A000045, A022629, A300520, A318263, A318264.

nmax = 50; CoefficientList[Series[Product[1 + Fibonacci[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += Fibonacci[k]*poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

From Vaclav Kotesovec, Aug 24 2018: (Start)
a(n) ~ c * A000045(n) * exp(r*sqrt(n)) / n^(3/4) ~ c * exp(r*sqrt(n)) * phi^n / (sqrt(5) * n^(3/4)), where r = 2*sqrt(-polylog(2, -1/sqrt(5))) = 1.273105657580344020952907652385896290122122879833..., c = 0.4521555113342405268628694407039776... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
Equivalently, r = 2*sqrt(Pi^2/6 + log(5)^2/8 + polylog(2, -sqrt(5))). (End)

A318264 Expansion of Product_{k>=1} (1 + C(k)*x^k), where C(k) is the Catalan number A000108.

Original entry on oeis.org

1, 1, 2, 7, 19, 66, 212, 743, 2487, 9012, 31177, 113775, 404584, 1490726, 5376676, 20028981, 73068861, 273659672, 1009921813, 3801386137, 14125670266, 53477758556, 199950414035, 759566205693, 2857261603610, 10889590477287, 41136917417501, 157329747348492

Offset: 0

Vaclav Kotesovec, Aug 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1650

Cf. A000108, A022629, A179381, A318248, A318263.

C:= proc(n) option remember; binomial(n+n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2019

nmax = 40; CoefficientList[Series[Product[1+CatalanNumber[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += CatalanNumber[k]*poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

a(n) ~ c * A000108(n) ~ c * 4^n / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + C(k)/4^k) = 2.608465265690846547082817204714986077801494... - Vaclav Kotesovec, Aug 24 2018

A325515 Sum of sums of omegas of the parts over all strict integer partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 11, 14, 22, 29, 37, 50, 63, 81, 106, 129, 160, 203, 246, 303, 373, 449, 541, 654, 782, 932, 1116, 1322, 1559, 1848, 2167, 2537, 2978, 3470, 4041, 4706, 5449, 6303, 7291, 8402, 9665, 11117, 12744, 14592, 16708, 19062, 21730, 24757, 28141

Offset: 0

Gus Wiseman, May 07 2019

nonn

Also omega of the product of products of parts over all strict integer partitions of n.

The omega of n is A001222(n), the number of prime factors of n counted with multiplicity.

Cf. A001222, A003963, A015716, A015723, A022629, A056239, A066189, A112798, A147655, A246867, A325504, A325506.

Table[Sum[Total[PrimeOmega/@s],{s,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

a(n) = A001222(A325504(n)).

A032309 "EFK" (unordered, size, unlabeled) transform of 2,4,6,8,...

Original entry on oeis.org

1, 2, 4, 14, 20, 50, 112, 190, 328, 666, 1340, 2038, 3740, 5954, 10792, 19542, 30048, 48290, 80164, 124694, 204484, 347610, 515184, 810750, 1240296, 1932722, 2887820, 4557838, 7126652, 10463330, 15768168, 23499934

Offset: 0

Christian G. Bower

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
C. G. Bower, Transforms (2)

Cf. A022629, A032302, A265951, A265955.

nmax=40; CoefficientList[Series[Product[(1+2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 19 2015 *)

seq(n)={Vec(prod(k=1, n, 1 + 2*k*x^k + O(x*x^n)))} \\ Andrew Howroyd, Sep 20 2018

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

a(0)=1 prepended by Andrew Howroyd, Sep 20 2018

A265955 Expansion of Product_{k>=1} (1 + 2kx^k)/(1 - 2kx^k).

Original entry on oeis.org

1, 4, 16, 60, 192, 596, 1776, 5020, 13760, 36916, 96336, 246316, 619392, 1530548, 3729392, 8976364, 21337920, 50195268, 116977232, 270114764, 618712640, 1406843940, 3176387120, 7126185948, 15894370816, 35253947940, 77796242768, 170868178332, 373606888128

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3280

Cf. A006906, A022629, A032302, A032309, A070933, A265951.

nmax=40; CoefficientList[Series[Product[(1+2*k*x^k)/(1-2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * n * 2^n, where c = 2 * Product_{m>=3} (1 + 2/(2^(m-1)/m - 1)) = 193.4198278838721371054040810054045645734538119720773785523616944906739...

A267008 Expansion of Product_{k>=1} (1 + (k+1)*x^k).

Original entry on oeis.org

1, 2, 3, 10, 13, 28, 58, 90, 146, 260, 481, 688, 1168, 1748, 2863, 4726, 6938, 10412, 16140, 23746, 35702, 55812, 79032, 116758, 168779, 247006, 350310, 513410, 744286, 1045466, 1485685, 2098780, 2935416, 4137878, 5746618, 8027612, 11343706, 15487222, 21418682

Offset: 0

Vaclav Kotesovec, Jan 08 2016

nonn

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

Cf. A022629, A074141, A267007.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 15 2019

nmax = 50; CoefficientList[Series[Product[1+(k+1)*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 2; Do[Do[poly[[j+1]] += (k+1)*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

A285240 Expansion of Product_{k>=1} (1 + k*x^k)^(k^2).

Original entry on oeis.org

1, 1, 8, 35, 115, 429, 1425, 4803, 15398, 48940, 151046, 459000, 1373219, 4037721, 11723911, 33566828, 94993571, 265722551, 735543433, 2015558930, 5471271099, 14719853084, 39266487114, 103908002173, 272855152096, 711272144097, 1841162650896, 4734074846631

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A022629, A027998, A266891, A285241, A285242.

nmax = 40; CoefficientList[Series[Product[(1 + k*x^k)^(k^2), {k,1,nmax}], {x,0,nmax}], x]
nmax = 40; s = 1 + x; Do[s*=Sum[Binomial[k^2, j]*k^j*x^(j*k), {j, 0, Floor[nmax/k] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x]

A302831 Expansion of (1/(1 - x))Product_{k>=1} (1 + kx^k).

Original entry on oeis.org

1, 2, 4, 9, 16, 31, 56, 99, 163, 283, 469, 757, 1220, 1915, 3020, 4748, 7273, 11014, 16789, 25033, 37480, 55782, 82206, 120033, 174762, 253092, 364276, 523814, 749438, 1064853, 1509561, 2128227, 2986392, 4186093, 5832169, 8121130, 11272081, 15576076, 21446615, 29479186, 40360980

Offset: 0

Ilya Gutkovskiy, Apr 13 2018

nonn

Partial sums of A022629.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A000009, A022629, A036469, A302830, A302832.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, i*b(n-i, i-1))))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 13 2018

nmax = 40; CoefficientList[Series[1/(1 - x) Product[(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[(-1)^(j + 1) k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]

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

A306919 Sum over all partitions of n into distinct parts of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in increasing order.

Original entry on oeis.org

1, 1, 2, 4, 5, 14, 24, 122, 318, 2417851639229258349414245, 14134776518227074636666380005943348126619871175004951664972849610340964762

Offset: 0

Alois P. Heinz, Mar 16 2019

nonn

			a(0) = 1 because the empty partition () has no parts, the exponentiation operator ^ is right-associative, and 1 is the right identity of exponentiation.
a(6) = 1^2^3 + 2^4 + 1^5 + 6 = 1 + 16 + 1 + 6 = 24.

Alois P. Heinz, Table of n, a(n) for n = 0..11
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Exponentiation
Wikipedia, Identity element
Wikipedia, Operator associativity
Wikipedia, Partition (number theory)

Cf. A022629, A066189, A306895, A306918.

d:= proc(l) local i; for i to nops(l)-1 do
       if l[i]=l[i+1] then return fi od; l
    end:
f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
a:= n-> add(f(l), l=map(l->d(sort(l, `<`)), combinat[partition](n))):
seq(a(n), n=0..11);

d[l_] := Module[{i}, For[i = 1, i <= Length[l]-1 , i++, If[l[[i]] == l[[i+1]], Return[]]]; l];
f[l_] := If[l == {}, 1, l[[1]]^f[Delete[l, 1]]];
a[n_] := Sum[f[l], {l, Sort /@ Select[IntegerPartitions[n], Length@# == Length @ Union@#&]}];
a /@ Range[0, 11] (* Jean-François Alcover, May 03 2020, after Maple *)

cp's OEIS Frontend

A303188 a(n) = [x^n] Product_{k=1..n} (1 + (n - k + 1)*x^k).

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A318248 Expansion of Product_{k>=1} (1 + Fibonacci(k)*x^k).

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A318264 Expansion of Product_{k>=1} (1 + C(k)*x^k), where C(k) is the Catalan number A000108.

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A325515 Sum of sums of omegas of the parts over all strict integer partitions of n.

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A032309 "EFK" (unordered, size, unlabeled) transform of 2,4,6,8,...

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PARI

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A265955 Expansion of Product_{k>=1} (1 + 2*k*x^k)/(1 - 2*k*x^k).

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Mathematica

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A267008 Expansion of Product_{k>=1} (1 + (k+1)*x^k).

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Maple

Mathematica

A285240 Expansion of Product_{k>=1} (1 + k*x^k)^(k^2).

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Mathematica

A302831 Expansion of (1/(1 - x))*Product_{k>=1} (1 + k*x^k).

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A265955 Expansion of Product_{k>=1} (1 + 2kx^k)/(1 - 2kx^k).

A302831 Expansion of (1/(1 - x))Product_{k>=1} (1 + kx^k).