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.

Showing 1-10 of 12 results. Next

A101509 Binomial transform of tau(n) (see A000005).

Original entry on oeis.org

1, 3, 7, 16, 35, 75, 159, 334, 696, 1442, 2976, 6123, 12562, 25706, 52492, 107014, 217877, 443061, 899957, 1826078, 3701783, 7498261, 15178255, 30706320, 62085915, 125465715, 253415981, 511608490, 1032427637, 2082680887, 4199956101, 8467124805, 17064784905, 34382825363, 69256687719, 139465867773
Offset: 0

Views

Author

Paul Barry, Dec 05 2004

Keywords

Comments

Row sums of A101508.
Also: Number of matrices with positive integer coefficients such that the sum of all entries equals n+1, cf. link "Partitions and A101509". - M. F. Hasler, Jan 14 2009

Examples

			From _Gus Wiseman_, Jan 16 2019: (Start)
The a(3) = 16 ways to arrange the parts of an integer partition of 4 into a matrix:
  [4] [1 3] [3 1] [2 2] [1 1 2] [1 2 1] [2 1 1] [1 1 1 1]
.
  [1] [3] [2] [1 1]
  [3] [1] [2] [1 1]
.
  [1] [1] [2]
  [1] [2] [1]
  [2] [1] [1]
.
  [1]
  [1]
  [1]
  [1]
(End)

Links

Crossrefs

Cf. A000005 (tau), A101508, A160399.
Cf. A000219, A047966, A053529, A319066, A323300, A323301, A323307, A323429.

Programs

  • Maple
    bintr:= proc(p) proc(n) add(p(k) *binomial(n, k), k=0..n) end end:
a:= bintr(n-> numtheory[tau](n+1)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 30 2011
  • Mathematica
    a[n_] := Sum[DivisorSigma[0, k+1]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2017 *)
  • PARI
    A101509(n) = sum( k=0,n, numdiv(k+1)*binomial(n,k)) \\ M. F. Hasler, Jan 14 2009

Formula

a(n) = Sum_{k=0..n, Sum_{i=0..n, if(mod(i+1, k+1)=0, binomial(n, i), 0)}}.
G.f.: 1/x * Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1-x). - Joerg Arndt, Jan 30 2011
a(n) ~ 2^n * (log(n/2) + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 07 2020

A185003 a(n) = Sum_{k=1..n} binomial(n,k)*sigma(k).

Original entry on oeis.org

1, 5, 16, 45, 116, 284, 673, 1557, 3535, 7910, 17502, 38376, 83500, 180479, 387881, 829605, 1766998, 3749765, 7931114, 16724870, 35173777, 73794660, 154485527, 322771344, 673155141, 1401536934, 2913490375, 6047714599, 12536770558, 25956242579, 53678385266
Offset: 1

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Examples

			L.g.f.: L(x) = x + 5*x^2/2 + 16*x^3/3 + 45*x^4/4 + 116*x^5/5 + 284*x^6/6 +...
where exponentiation yields A103446 (with offset=0):
exp(L(x)) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...

Links

Crossrefs

Cf. A103446, A222115, A000203, A160399.

Programs

  • Magma
    [&+[Binomial(n,k)*DivisorSigma(1,k):k in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019
  • Magma
    [&+[&+[i*Binomial(n,i*j):j in [1..n]]:i in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019
  • Maple
    with(numtheory): seq(add(binomial(n,i)*sigma(i), i=1..n), n=1..40); # Ridouane Oudra, Nov 12 2019
  • Mathematica
    Table[Sum[Binomial[n, k] DivisorSigma[1, k], {k, n}], {n,50}] (* G. C. Greubel, Jun 03 2017 *)
  • PARI
    {a(n)=sum(k=1,n,sigma(k)*binomial(n,k))}
for(n=1,30,print1(a(n),", "))
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(m=1, n+1, x^m/((1-x)^m-X^m)/m), n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(k=1, n, k*log(1-X)-log((1-x)^k-X^k)), n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m), n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(k=1, n, valuation(2*k, 2)*log(1 + x^k/(1-X)^k)), n)}

Formula

Logarithmic derivative of A103446 (with offset=0), which describes the binomial transform of partitions.
From Paul D. Hanna, Jun 01 2013: (Start)
L.g.f.: Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n.
L.g.f.: Sum_{n>=1} x^n/((1-x)^n - x^n) / n.
L.g.f.: Sum_{n>=1} n*log(1-x) - log((1-x)^n - x^n).
L.g.f.: Sum_{n>=1} A001511(n) * log(1 + x^n/(1-x)^n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
a(n) = A222115(n) - 1. (End)
a(n) ~ Pi^2/12 * n * 2^n. - Vaclav Kotesovec, Dec 30 2015
a(n) = Sum_{i=1..n} Sum_{j=1..n} i*binomial(n,i*j). - Ridouane Oudra, Nov 12 2019

A295739 Expansion of e.g.f. exp(Sum_{k>=1} d(k)*x^k/k!), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 1, 3, 9, 36, 158, 802, 4434, 26978, 176637, 1243528, 9316519, 74065506, 621187700, 5480130494, 50662481722, 489552042241, 4931215686119, 51668848043427, 561981734692781, 6333882472789914, 73850048237680936, 889461218944314524, 11051067390893340510
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 26 2017

Keywords

Comments

Exponential transform of A000005.

Links

Crossrefs

Cf. A000005, A006171, A028342, A038200, A129921, A160399, A274804, A294363.

Programs

  • Maple
    a:=series(exp(add(tau(k)*x^k/k!,k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] DivisorSigma[0, k] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

Formula

E.g.f.: exp(Sum_{k>=1} A000005(k)*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A000005(k)*a(n-k).

A306988 a(n) = Sum_{k=1..n} binomial(n,k)*phi(k), where phi is the Euler totient function.

Original entry on oeis.org

1, 3, 8, 20, 49, 117, 272, 620, 1395, 3107, 6852, 14964, 32395, 69647, 149002, 317712, 675749, 1433769, 3033444, 6396320, 13437913, 28130869, 58708304, 122239396, 254141275, 527946013, 1096312050, 2275897660, 4722500707, 9791471587, 20277706762, 41932520528
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 18 2019

Keywords

Links

Crossrefs

Cf. A000010, A002088, A101509, A160399, A185003.
Partial sums of A131045.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*EulerPhi[k], {k, 1, n}], {n, 1, 40}]

Formula

a(n) ~ 3 * n * 2^n / Pi^2.

A356038 a(n) = Sum_{k=1..n} binomial(n,k) * sigma_2(k).

Original entry on oeis.org

1, 7, 28, 95, 286, 802, 2143, 5519, 13807, 33762, 81060, 191678, 447396, 1032647, 2360593, 5351231, 12041764, 26920297, 59829006, 132262550, 290990077, 637429514, 1390811841, 3023647046, 6551547161, 14151910442, 30481920523, 65480947739, 140318385088, 299995596747
Offset: 1

Views

Author

Vaclav Kotesovec, Jul 24 2022

Keywords

Links

Crossrefs

Cf. A001157, A160399, A185003, A356039.
Cf. A006218, A024916, A064602, A064603.

Programs

  • Maple
    with(numtheory): seq(add(sigma[2](i)*binomial(n,i),i=1..n), n=1..60); # Ridouane Oudra, Oct 25 2022
  • Mathematica
    Table[Sum[Binomial[n, k] * DivisorSigma[2, k], {k, 1, n}], {n, 1, 40}]
  • PARI
    a(n) = sum(k=1, n, binomial(n,k) * sigma(k, 2)); \\ Michel Marcus, Jul 24 2022

Formula

a(n) ~ zeta(3) * n^2 * 2^(n-2).
a(n) = Sum_{i=1..n} Sum_{j=1..n} (i^2)*binomial(n,i*j). - Ridouane Oudra, Oct 25 2022

A308554 Expansion of e.g.f. Sum_{k>=1} tau(k)*(exp(x) - 1)^k/k!, where tau = number of divisors (A000005).

Original entry on oeis.org

1, 3, 9, 30, 113, 472, 2145, 10514, 55428, 313255, 1886888, 12029741, 80701715, 567541878, 4175795147, 32104799401, 257561662496, 2151841672173, 18676002357864, 167951667633495, 1561420657033927, 14980472336450530, 148140814019762129, 1508776236781766431
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 07 2019

Keywords

Comments

Stirling transform of A000005.

Links

Crossrefs

Cf. A000005, A160399, A308555.

Programs

  • Maple
    b:= proc(n, m) option remember; uses numtheory;
     `if`(n=0, tau(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..24);  # Alois P. Heinz, Aug 04 2021
  • Mathematica
    nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[0, k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[StirlingS2[n, k] DivisorSigma[0, k], {k, 1, n}], {n, 1, 24}]

Formula

G.f.: Sum_{k>=1} tau(k)*x^k / Product_{j=1..k} (1 - j*x).
a(n) = Sum_{k=1..n} Stirling2(n,k)*tau(k).

A324914 a(n) = Sum_{k=1..n} 2^k * tau(k), where tau(k) = A000005(k).

Original entry on oeis.org

2, 10, 26, 74, 138, 394, 650, 1674, 3210, 7306, 11402, 35978, 52362, 117898, 248970, 576650, 838794, 2411658, 3460234, 9751690, 18140298, 34917514, 51694730, 185912458, 286575754, 555011210, 1091882122, 2702494858, 3776236682, 12366171274
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 18 2019

Keywords

Comments

Partial sums of A323351 with n=0 term of A323351 omitted. - Robert Israel, Jun 27 2019

Links

Crossrefs

Cf. A000005, A006218, A101509, A160399, A324915, A323351.

Programs

  • Maple
    ListTools:-PartialSums([seq(2^k*numtheory:-tau(k),k=1..100)]); # Robert Israel, Jun 27 2019
  • Mathematica
    Accumulate[Table[2^k*DivisorSigma[0, k], {k, 1, 30}]]

A328681 a(n) = Sum_{k=1..n} binomial(n,k) * tau(k) * tau(n - k + 1), where tau = A000005.

Original entry on oeis.org

1, 6, 20, 55, 142, 322, 779, 1608, 3894, 7370, 18372, 33137, 81512, 149694, 353224, 641461, 1570836, 2684928, 6642915, 11795178, 28133846, 46768200, 125433400, 197654545, 485749918, 893864394, 2066417482, 3385115393, 8975476976, 14384181908, 35478028091, 61940000322
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 03 2019

Keywords

Crossrefs

Cf. A000005, A007318, A055507, A160399, A330088.

Programs

  • Magma
    [&+[Binomial(n,k)*DivisorSigma(0,k)*DivisorSigma(0,n-k+1):k in [1..n]]:n in [1..32]]; // Marius A. Burtea, Dec 03 2019
  • Mathematica
    Table[Sum[Binomial[n, k] DivisorSigma[0, k] DivisorSigma[0, n - k + 1], {k, 1, n}], {n, 1, 32}]
nmax = 32; CoefficientList[Series[(1/2) D[Sum[DivisorSigma[0, k] x^k/k!, {k, 1, nmax}]^2, x], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n) = sum(k=1, n, binomial(n,k)*numdiv(k)*numdiv(n-k+1)); \\ Michel Marcus, Dec 05 2019

Formula

E.g.f.: (1/2) * d/dx (Sum_{k>=1} tau(k) * x^k / k!)^2.

A307679 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^(1/k).

Original entry on oeis.org

1, 1, 5, 35, 323, 3679, 49819, 781465, 13923545, 277563617, 6118251461, 147715469131, 3875706370315, 109781717161375, 3338229675519803, 108443658227589329, 3747688533281296049, 137273241169036231105, 5311844045472206624005, 216505267421266611639667, 9270689769095765333645651
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 21 2019

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 323*x^4/4! + 3679*x^5/5! + 49819*x^6/6! + 781465*x^7/7! + 13923545*x^8/8! + ...
log(A(x)) = x + 4*x^2/2 + 11*x^3/3 + 27*x^4/4 + 62*x^5/5 + 137*x^6/6 + 296*x^7/7 + 630*x^8/8 + 1326*x^9/9 + ... + A160399(k)*x^k/k + ...

Crossrefs

Cf. A000005, A028342, A103446, A160399, A218482, A307680, A320563.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, 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} d(k)*x^k/(k*(1 - x)^k)), where d(k) is the number of divisors of k (A000005).
a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A028342(k)*n!/k!.

A356039 a(n) = Sum_{k=1..n} binomial(n,k) * sigma_3(k).

Original entry on oeis.org

1, 11, 58, 243, 866, 2804, 8485, 24387, 67333, 180086, 469338, 1196976, 2996956, 7385837, 17954243, 43125267, 102494548, 241309031, 563341508, 1305142418, 3002938045, 6866090880, 15609292379, 35299794600, 79443050541, 177989130174, 397124963671, 882642816697, 1954708794400
Offset: 1

Views

Author

Vaclav Kotesovec, Jul 24 2022

Keywords

Comments

For m>0, Sum_{k=1..n} binomial(n,k) * sigma_m(k) ~ zeta(m+1) * n^m * 2^(n-m).

Links

Crossrefs

Cf. A001158, A160399, A185003, A356038.
Cf. A006218, A024916, A064602, A064603.

Programs

  • Maple
    with(numtheory): seq(add(sigma[3](i)*binomial(n,i), i=1..n), n=1..60); # Ridouane Oudra, Oct 31 2022
  • Mathematica
    Table[Sum[Binomial[n, k] * DivisorSigma[3, k], {k, 1, n}], {n, 1, 40}]
  • PARI
    a(n) = sum(k=1, n, binomial(n,k) * sigma(k, 3)); \\ Michel Marcus, Jul 24 2022

Formula

a(n) ~ Pi^4 * n^3 * 2^(n-4) / 45.
a(n) = Sum_{i=1..n} Sum_{j=1..n} (i^3)*binomial(n,i*j). - Ridouane Oudra, Oct 31 2022
Showing 1-10 of 12 results. Next

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