cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 20 results.

A318570 Expansion of Product_{k>=1} ((1 - x)^k + x^k)/((1 - x)^k - x^k).

Original entry on oeis.org

1, 2, 6, 18, 52, 146, 402, 1090, 2916, 7708, 20160, 52236, 134222, 342304, 867024, 2182384, 5461696, 13595918, 33677550, 83036878, 203859820, 498470998, 1214230586, 2947204870, 7129403128, 17191258642, 41328057106, 99067295658, 236822823336, 564650823162, 1342921372126
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 15 2018

Keywords

Comments

First differences of the binomial transform of A015128.
Convolution of A129519 and A218482.

Links

Crossrefs

Cf. A000203, A015128, A129519, A218482, A266497.

Programs

  • Maple
    a:=series(mul(((1-x)^k+x^k)/((1-x)^k-x^k),k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 - x)^k + x^k)/((1 - x)^k - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[1/EllipticTheta[4, 0, x/(1 - x)], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: 1/theta_4(x/(1 - x)), where theta_4() is the Jacobi theta function.
G.f.: exp(Sum_{k>=1} (sigma(2*k) - sigma(k))*x^k/(k*(1 - x)^k)).
a(n) ~ 2^(n-3) * exp(Pi*sqrt(n/2) + Pi^2/16) / n. - Vaclav Kotesovec, Oct 15 2018

A380412 First term of the n-th differences of the strict partition numbers. Inverse zero-based binomial transform of A000009.

Original entry on oeis.org

1, 0, 0, 1, -3, 7, -14, 25, -41, 64, -100, 165, -294, 550, -1023, 1795, -2823, 3658, -2882, -2873, 20435, -62185, 148863, -314008, 613957, -1155794, 2175823, -4244026, 8753538, -19006490, 42471787, -95234575, 210395407, -453413866, 949508390, -1931939460
Offset: 0

Views

Author

Gus Wiseman, Feb 03 2025

Keywords

Comments

Up to sign, same as A293467.

Crossrefs

The version for non-strict partitions is A281425, row n=0 of A175804.
Column n=0 of A378622.
A000009 counts strict integer partitions, differences A087897, A378972.
A266232 gives zero-based binomial transform of A000009, differences A129519.
Cf. A000041, A007442, A008284, A218482, A293467, A320590, A320591, A377285, A378970.

Programs

  • Mathematica
    nn=10;Table[First[Differences[PartitionsQ/@Range[0,nn],n]],{n,0,nn}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k) A000041(k).

A307311 Expansion of Product_{k>=1} 1/(1 + x^k/(1 - x)^k).

Original entry on oeis.org

1, -1, -1, -2, -3, -4, -5, -6, -6, -1, 19, 74, 200, 461, 977, 1987, 3976, 7902, 15559, 30105, 56778, 103833, 183765, 314882, 523007, 841752, 1305431, 1916607, 2540433, 2609983, 381628, -8814988, -36463325, -109113400, -285322360, -689608522, -1579574566, -3477967848
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 02 2019

Keywords

Comments

First differences of the binomial transform of A081362.
Convolution inverse of A129519.

Links

Crossrefs

Cf. A081362, A129519, A218482, A307310.

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(&*[(1+x^k/(1-x)^k): k in [1..m+2]]) )); // G. C. Greubel, Apr 03 2019
  • Maple
    a:=series(mul(1/(1+x^k/(1-x)^k),k=1..100), x=0,38): seq(coeff(a, x, n), n=0..37); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 + x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    m=40; my(x='x+O('x^m)); Vec( 1/prod(k=1,m+2, (1+x^k/(1-x)^k)) ) \\ G. C. Greubel, Apr 03 2019
  • Sage
    m=40; (1/product(1+x^k/(1-x)^k for k in (1..m+2))).series(x, m).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019

A307548 Expansion of Product_{k>=1} (1 - (x/(1+x))^k).

Original entry on oeis.org

1, -1, 0, 1, -2, 4, -9, 21, -48, 105, -218, 429, -803, 1442, -2521, 4380, -7734, 14091, -26468, 50405, -94980, 172824, -296704, 467589, -644459, 678109, -177123, -1752141, 7003180, -19432494, 46778567, -104623822, 224830880, -473859273, 992825436, -2084921584
Offset: 0

Views

Author

Seiichi Manyama, Apr 14 2019

Keywords

Crossrefs

Convolution inverse of A320590.
Cf. A129519, A307310, A320591.

Programs

  • Mathematica
    m = 35; CoefficientList[Series[Product[1 - (x/(1+x))^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-(x/(1+x))^k))

Formula

O.g.f.: Sum_{n >= 0} (-1)^n * x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A320591. - Peter Bala, Dec 22 2020

A320586 Expansion of (1/(1 - x)) * Sum_{k>=1} k*x^k/(x^k + (1 - x)^k).

Original entry on oeis.org

1, 3, 10, 27, 66, 156, 365, 843, 1909, 4238, 9274, 20136, 43564, 94013, 202155, 432475, 919820, 1945767, 4098852, 8610922, 18061277, 37844128, 79212323, 165565920, 345412341, 719047566, 1493488927, 3095654281, 6405734456, 13238611241, 27336762272, 56416256443, 116376652600
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 16 2018

Keywords

Comments

Binomial transform of A000593.

Links

Crossrefs

Cf. A000593, A129519, A185003, A320589.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1/(1 +-x))*(&+[k*x^k/(x^k + (1 - x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 30 2018
  • Maple
    seq(coeff(series((1/(1-x))*add(k*x^k/(x^k+(1-x)^k),k=1..n),x,n+1), x, n), n = 1 .. 35); # Muniru A Asiru, Oct 16 2018
  • Mathematica
    nmax = 33; Rest[CoefficientList[Series[1/(1 - x) Sum[k x^k/(x^k + (1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 33; Rest[CoefficientList[Series[(EllipticTheta[3, 0, x/(1 - x)]^4 + EllipticTheta[2, 0, x/(1 - x)]^4 - 1)/(24 (1 - x)), {x, 0, nmax}], x]]
Table[Sum[Binomial[n, k] Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}], {k, n}], {n, 33}]
  • PARI
    m=50; x='x+O('x^m); Vec((1/(1 - x))*sum(k=1, m+2, k*x^k/(x^k + (1 - x)^k))) \\ G. C. Greubel, Oct 30 2018

Formula

G.f.: (theta_3(x/(1 - x))^4 + theta_2(x/(1 - x))^4 - 1)/(24*(1 - x)), where theta_() is the Jacobi theta function.
L.g.f.: Sum_{k>=1} A000593(k)*x^k/(k*(1 - x)^k) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{k=1..n} binomial(n,k)*A000593(k).
Conjecture: a(n) ~ c * 2^n * n, where c = Pi^2/24 = 0.411233516712... - Vaclav Kotesovec, Jun 26 2019

A307262 Expansion of Product_{k>=1} (1 + k*x^k/(1 - x)^k).

Original entry on oeis.org

1, 1, 3, 10, 29, 82, 231, 646, 1780, 4835, 13009, 34794, 92600, 245119, 644983, 1686869, 4387030, 11353686, 29261059, 75134965, 192261744, 490305251, 1246128051, 3156425284, 7969135647, 20057905672, 50339682075, 126002008265, 314604617989, 783668652379, 1947689149020
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

First differences of the binomial transform of A022629.

Crossrefs

Cf. A022629, A129519, A307259, A307261, A320564.

Programs

  • Maple
    a:=series(mul(1+k*x^k/(1-x)^k,k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 + k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

A307680 Expansion of e.g.f. Product_{k>=1} (1 + x^k/(1 - x)^k)^(1/k).

Original entry on oeis.org

1, 1, 3, 17, 131, 1239, 14029, 187627, 2906553, 50982929, 993806531, 21270277401, 496425262123, 12577053063847, 344382608381421, 10139294386051139, 319175215666010609, 10684742192933940897, 378662321114852778883, 14158327369578651838369, 557151639159864934384851
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 21 2019

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 131*x^4/4! + 1239*x^5/5! + 14029*x^6/6! + 187627*x^7/7! + 2906553*x^8/8! + ...

Crossrefs

Cf. A048272, A129519, A168243, A307679, A320564.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + x^k/(1 - x)^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[(-1)^(d + 1), {d, Divisors[k]}] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(Sum_{k>=1} A048272(k)*x^k/(k*(1 - x)^k)).
a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A168243(k)*n!/k!.

A332024 E.g.f.: Product_{k>=1} (1 + x^k/(k!*(1 - x)^k)).

Original entry on oeis.org

1, 1, 3, 16, 113, 956, 9382, 105253, 1334517, 18904936, 295787126, 5056826039, 93594929738, 1861321879535, 39536014577711, 892763601542509, 21352130132268541, 539243894127067888, 14342761454293102006, 400830115867761118963, 11743833994363640228070
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 13 2020

Keywords

Crossrefs

Cf. A007837, A084357, A129519, A320567, A330649.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + x^k/(k! (1 - x)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    seq(n)={Vec(serlaplace(prod(k=1, n, (1 + x^k/(k!*(1 - x)^k)) + O(x*x^n))))} \\ Andrew Howroyd, Feb 13 2020

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * A007837(k) * n! / k!.

A352120 G.f. A(x) satisfies: Product_{n>=1} (1 + x^n*A(x)) = Product_{n>=1} (1 + x^n/(1-x)^n).

Original entry on oeis.org

1, 1, 2, 5, 10, 20, 43, 93, 194, 403, 842, 1755, 3656, 7643, 15976, 33281, 69164, 143558, 297619, 616625, 1277729, 2647861, 5485300, 11356731, 23495794, 48567063, 100301668, 206994479, 426941231, 880227976, 1814221503, 3738368348, 7701376466
Offset: 0

Views

Author

Paul D. Hanna, Mar 05 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 20*x^5 + 43*x^6 + 93*x^7 + 194*x^8 + 403*x^9 + 842*x^10 + 1755*x^11 + 3656*x^12 + ...
such that the following products are equal:
P(x) = (1 + x*A(x)) * (1 + x^2*A(x)) * (1 + x^3*A(x)) * (1 + x^4*A(x)) * (1 + x^5*A(x)) * (1 + x^6*A(x)) * ...
P(x) = (1 + x/(1-x)) * (1 + x^2/(1-x)^2) * (1 + x^3/(1-x)^3) * (1 + x^4/(1-x)^4) * (1 + x^5/(1-x)^5) * ...
also, we have the sums
P(x) = 1 + x*A(x)/(1-x) + x^3*A(x)^2/((1-x)*(1-x^2)) + x^6*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^10*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + ...
1/P(x) = 1 - x*A(x)/(1-x) + x^2*A(x)^2/((1-x)*(1-x^2)) - x^3*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^4*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+ ...
where
P(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 28*x^5 + 65*x^6 + 151*x^7 + 350*x^8 + 807*x^9 + 1850*x^10 + ... + A129519(n)*x^n + ...

Links

Crossrefs

Cf. A129519.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( prod(n=1,#A, (1 + x^n/(1-x +x*O(x^#A))^n)/(1 + x^n*Ser(A)) ),#A) );A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n and P(x) = Product_{n>=1} (1 + x^n/(1-x)^n) satisfies:
(1) P(x) = Product_{n>=1} (1 + x^n*A(x)).
(2) P(x) = Sum_{n>=0} x^(n*(n+1)/2) * A(x)^n / (Product_{k=1..n} (1 - x^k)).
(3) 1/P(x) = Sum_{n>=0} (-x)^n * A(x)^n / (Product_{k=1..n} (1 - x^k)).
(4) log(P(x)) = Sum_{n>=1} x^n * Sum_{d|n} -(-A(x))^(n/d) * d/n.

A307265 Expansion of Product_{k>=1} 1/(1 + (-x)^k/(1 - x)^k).

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 27, 58, 122, 257, 549, 1190, 2600, 5683, 12367, 26749, 57530, 123202, 263115, 561131, 1196248, 2550975, 5443115, 11620526, 24814735, 52979512, 113038103, 240936717, 512916683, 1090501249, 2315608462, 4911611864, 10408318627, 22040127864
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

First differences of the binomial transform of A000700.

Crossrefs

Cf. A000700, A129519, A218482, A307264.

Programs

  • Maple
    a:=series(mul(1/(1+(-x)^k/(1-x)^k),k=1..50),x=0,34): seq(coeff(a,x,n),n=0..33); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 33; CoefficientList[Series[Product[1/(1 + (-x)^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^(2*k-1)/(1 - x)^(2*k-1)).
a(n) ~ 2^(n-2) * exp(Pi*sqrt(n/3)/2 + Pi^2/96) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 01 2019
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