A185003 -id:A185003 - cp's OEIS frontend

A160399 a(n) = Sum_{k=1..n} binomial(n,k) * d(k), where d(k) = the number of positive divisors of k.

1, 4, 11, 27, 62, 137, 296, 630, 1326, 2768, 5744, 11867, 24429, 50135, 102627, 209641, 427518, 870579, 1770536, 3596614, 7298397, 14796658, 29974913, 60681233, 122767148, 248232863, 501648844, 1013257334, 2045684971

Offset: 1

Leroy Quet, May 12 2009

nonn

Binomial transform of the sequence d(n) (A000005). - Emeric Deutsch, May 15 2009

Apparently the partial sums of A101509. - R. J. Mathar, May 17 2009

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric M. Schmidt, The probability that the number of points on a complete intersection is squarefree, Rocky Mountain J. Math, Volume 47, Number 8 (2017), 2777-2796.

Cf. A000005. - Emeric Deutsch, May 15 2009

Cf. A101509, A185003.

List([1..10^3], n -> Sum([1..n], k -> Binomial(n,k) * Number(DivisorsInt(k)))); # Muniru A Asiru, Feb 04 2018

[&+[Binomial(n,k)*NumberOfDivisors(k):k in [1..n]]:n in [1..30]]; // Marius A. Burtea, Nov 12 2019

[&+[&+[Binomial(n,i*j):j in [1..n]]:i in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019

with(numtheory): seq(sum(binomial(n, k)*tau(k), k = 1 .. n), n = 1 .. 30); # Emeric Deutsch, May 15 2009
A160399 := proc(n) local k; add(binomial(n,k)*numtheory[tau](k),k=1..n) ; end: seq(A160399(n),n=1..40) ; # R. J. Mathar, May 17 2009

a[n_] := Sum[Binomial[n, k]*DivisorSigma[0, k], {k, 1, n}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 25 2017 *)

a(n) = sum(k=1, n, binomial(n, k)*numdiv(k)); \\ Michel Marcus, Feb 25 2017

G.f.: (Sum_{k>=1} (x/(1-x))^k/(1-x^k/(1-x)^k))/(1-x). - Emeric Deutsch, May 15 2009
E.g.f.: exp(x)*Sum_{k>=1} d(k)*x^k/k!. - Ilya Gutkovskiy, Nov 26 2017
a(n) = 2^n*(log(n) + 2*gamma - log(2)) + O(2^n*n^(-1/4)). [Put alpha_n = beta_n = 1/2 in Thm. 4.2 of Schmidt.] - Eric M. Schmidt, Feb 03 2018
a(n) = Sum_{i=1..n} Sum_{j=1..n} binomial(n,i*j). - Ridouane Oudra, Nov 12 2019

More terms from Emeric Deutsch and R. J. Mathar, May 15 2009

A103446 Unlabeled analog of A025168.

Original entry on oeis.org

0, 1, 3, 8, 21, 54, 137, 344, 856, 2113, 5179, 12614, 30548, 73595, 176455, 421215, 1001388, 2371678, 5597245, 13166069, 30873728, 72185937, 168313391, 391428622, 908058205, 2101629502, 4853215947, 11183551059, 25718677187, 59030344851, 135237134812

Offset: 0

Thomas Wieder, Feb 06 2005; revised Feb 20 2006

nonn

Or, if the initial 0 is omitted, this is the binomial transform of the partition numbers p(1), p(2), ... = 1, 2, 3, 5, 7, 11, 15, 22, 30, ... (A000041 without the initial 1).
The most precise definition of this sequence is the Maple combstruct command given below. See the first Wieder link for further details.
Sequence appears to have a rational o.g.f. - Ralf Stephan, May 18 2007
For n>0, row sums of triangle A137151. - Gary W. Adamson, Jan 23 2008
a(n) = A218482(n) for n>=1; see A218482 for more formulas.

			Let {} denote a set, [] a list and Z an unlabeled element.
a(3) = 8 because we have {[[Z]],[[Z]],[[Z]]}, {[[Z],[Z]],[[Z]]}, {[[Z],[Z],[Z]]}, {[[Z],[Z,Z]]}, {[[Z,Z],[Z]]}, {[[Z,Z]],[[Z]]}, {[[Z]],[[Z,Z]]}, {[[Z,Z,Z]]}.

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
Thomas Wieder, Expanded definitions of A103446 and A025168

Cf. A025168, A034691, A050351, A137151, A185003 (log), A218482.

with(combstruct); SubSetSeqU := [T,{T=Subst(U,S),S=Set(U,card>=1),U=Sequence(Z,card>=1)},unlabeled]; [seq(count(SubSetSeqU, size=n), n=0..30)];
allstructs(SubSetSeq,size=3); # to get the structures for n=3 - this output is shown in the example lines.

Flatten[{0, Table[Sum[Binomial[n-1,k]*PartitionsP[k+1],{k,0,n-1}],{n,1,30}]}] (* Vaclav Kotesovec, Jun 25 2015 *)

{a(n)=if(n<1,0,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x+x*O(x^n))^m/m)),n))} \\ Paul D. Hanna, Apr 21 2010

{a(n)=if(n<1,0,polcoeff(exp(sum(m=1,n,x^m/m*sum(k=1,m,binomial(m,k)*sigma(k)))+x*O(x^n)),n))} \\ Paul D. Hanna, Feb 04 2012

Vec(1/eta('x/(1-'x)+O('x^66))) \\ Joerg Arndt, Jul 30 2011

O.g.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x)^n/n ) - 1. - Paul D. Hanna, Apr 21 2010
O.g.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k)*sigma(k) ) - 1. - Paul D. Hanna, Feb 04 2012
O.g.f. P(x/(1-x)), where P(x) is the o.g.f. for number of partitions (A000041) a(n)=sum_{k=1,n} ( binomial(n-1,k-1)*A000041(k)). - Vladimir Kruchinin, Aug 10 2010
a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015

I can confirm that the terms shown are the binomial transform of the partition sequence 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, ... (A000041 without the a(0) term). - N. J. A. Sloane, May 18 2007

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

Vaclav Kotesovec, Mar 18 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..3280
Vaclav Kotesovec, Plot of a(n)/(n*2^n) for n = 1..5000
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Cf. A000010, A002088, A101509, A160399, A185003.

Partial sums of A131045.

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

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

A320568 a(n) = Sum_{k=1..n} (-1)^(n-k)binomial(n,k)sigma(k).

Original entry on oeis.org

1, 1, -2, 5, -14, 40, -111, 293, -731, 1726, -3882, 8408, -17796, 37423, -79337, 170917, -373812, 823585, -1809844, 3934750, -8424747, 17749392, -36874749, 75862016, -155359339, 318331170, -655146929, 1356952623, -2828151136, 5920984735, -12420140296, 26036563525

Offset: 1

Ilya Gutkovskiy, Oct 15 2018

sign

Inverse binomial transform of A000203.

Vaclav Kotesovec, Table of n, a(n) for n = 1..3000
Vaclav Kotesovec, Graph abs(a(n)) / (n*2^n)
N. J. A. Sloane, Transforms

Cf. A000203, A038200, A185003.

Flat(List([1..10],n->Sum([1..n],k->(-1)^(n-k)*Binomial(n,k)*Sigma(k))));  # Muniru A Asiru, Oct 15 2018

seq(add((-1)^(n-k)*binomial(n,k)*sigma(k),k=1..n),n=1..32); # Paolo P. Lava, Oct 25 2018

Table[Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[1, k], {k, n}], {n, 32}]
nmax = 32; Rest[CoefficientList[Series[1/(1 + x) Sum[k x^k/((1 + x)^k - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

G.f.: (1/(1 + x))*Sum_{k>=1} k*x^k/((1 + x)^k - x^k).
L.g.f.: Sum_{k>=1} sigma(k)*x^k/(k*(1 + x)^k) = Sum_{n>=1} a(n)*x^n/n.
Conjecture: a(n) ~ (-1)^n * Pi^2/48 * n * 2^n. - Vaclav Kotesovec, Oct 16 2018

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

Vaclav Kotesovec, Jul 24 2022

nonn

Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A001157, A160399, A185003, A356039.

Cf. A006218, A024916, A064602, A064603.

with(numtheory): seq(add(sigma[2](i)*binomial(n,i),i=1..n), n=1..60); # Ridouane Oudra, Oct 25 2022

Table[Sum[Binomial[n, k] * DivisorSigma[2, k], {k, 1, n}], {n, 1, 40}]

a(n) = sum(k=1, n, binomial(n,k) * sigma(k, 2)); \\ Michel Marcus, Jul 24 2022

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

A222115 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * sigma(k).

Original entry on oeis.org

2, 6, 17, 46, 117, 285, 674, 1558, 3536, 7911, 17503, 38377, 83501, 180480, 387882, 829606, 1766999, 3749766, 7931115, 16724871, 35173778, 73794661, 154485528, 322771345, 673155142, 1401536935, 2913490376, 6047714600, 12536770559, 25956242580, 53678385267, 110889844998

Offset: 1

Paul D. Hanna, Jun 01 2013

nonn

Here sigma(n) is the sum of divisors of n (A000203).

			L.g.f.: L(x) = 2*x + 6*x^2/2 + 17*x^3/3 + 46*x^4/4 + 117*x^5/5 + 285*x^6/6 +...
where
exp(L(x)) = 1 + 2*x + 5*x^2 + 13*x^3 + 34*x^4 + 88*x^5 + 225*x^6 + 569*x^7 +...+ A218481(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A218481, A185003, A000203.

Table[Sum[Binomial[n,k]DivisorSigma[1,k],{k,n}],{n,40}]+1 (* Harvey P. Dale, Jul 21 2015 *)

{a(n)=1+sum(k=1,n,binomial(n,k)*sigma(k))}
for(n=1,30,print1(a(n),", "))

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(m=1, n+1, x^m/((1-x)^m-X^m)/m), n)}

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(k=1, n, k*log(1-X)-log((1-x)^k-X^k)), n)}

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m), n)}

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(k=1, n, valuation(2*k, 2)*log(1 + x^k/(1-X)^k)), n)}

Logarithmic derivative of the binomial transform of the partition numbers (A218481).
L.g.f.: -log(1-x) + Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n.
L.g.f.: -log(1-x) + Sum_{n>=1} x^n/((1-x)^n - x^n) / n.
L.g.f.: -log(1-x) + Sum_{n>=1} n*log(1-x) - log((1-x)^n - x^n).
L.g.f.: -log(1-x) + 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) = A185003(n) + 1.
a(n) ~ Pi^2/12 * n * 2^n. - Vaclav Kotesovec, Dec 30 2015

A308555 Expansion of e.g.f. Sum_{k>=1} sigma(k)*(exp(x) - 1)^k/k!, where sigma = sum of divisors (A000203).

Original entry on oeis.org

1, 4, 14, 53, 222, 1011, 4944, 25884, 144963, 865556, 5477661, 36518635, 255323564, 1867122987, 14259709474, 113593734317, 942317654779, 8123227487723, 72599829900774, 671199117610868, 6407156027307909, 63061416571124056, 639303956718643041, 6670690645674913424

Offset: 1

Ilya Gutkovskiy, Jun 07 2019

nonn

Stirling transform of A000203.

Alois P. Heinz, Table of n, a(n) for n = 1..575

Cf. A000203, A185003, A308554.

b:= proc(n, m) option remember; uses numtheory;
     `if`(n=0, sigma(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 03 2021

nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[1, k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[StirlingS2[n, k] DivisorSigma[1, k], {k, 1, n}], {n, 1, 24}]

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

a(n) = Sum_{k=1..n} Stirling2(n,k)*sigma(k).

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

Ilya Gutkovskiy, Oct 16 2018

nonn

Binomial transform of A000593.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Plot of a(n)/(n*2^n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A000593, A129519, A185003, A320589.

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

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

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}]

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

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

A324915 a(n) = Sum_{k=1..n} 2^k * sigma(k), where sigma(k) = A000203(k).

Original entry on oeis.org

2, 14, 46, 158, 350, 1118, 2142, 5982, 12638, 31070, 55646, 170334, 285022, 678238, 1464670, 3496286, 5855582, 16079198, 26564958, 70605150, 137714014, 288708958, 490035550, 1496668510, 2536855902, 5355428190, 10724137310, 25756522846, 41862650206, 119172061534

Offset: 1

Vaclav Kotesovec, Mar 18 2019

nonn

Vaclav Kotesovec, Plot of a(n)/(n*2^n) for n = 1..100000
Eric Weisstein's World of Mathematics, Divisor Function

Cf. A000203, A024916, A185003, A324914.

Accumulate[Table[2^k*DivisorSigma[1, k], {k, 1, 30}]]

A330088 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(k) * sigma(n - k + 1), where sigma = A000203.

Original entry on oeis.org

1, 9, 43, 155, 511, 1442, 4131, 10323, 28171, 63987, 171667, 369395, 957958, 2047694, 5078963, 10671529, 26542339, 53522031, 132273403, 268623854, 647842889, 1266118858, 3197923083, 6058756355, 14581380971, 29480406552, 68634048862, 131847974143, 323289015466, 611887749996

Offset: 1

Ilya Gutkovskiy, Dec 03 2019

nonn

Cf. A000203, A000385, A007318, A185003, A328681.

[&+[Binomial(n,k)*DivisorSigma(1,k)*DivisorSigma(1,n-k+1):k in [1..n]]:n in [1..30]]; // Marius A. Burtea, Dec 03 2019

Table[Sum[Binomial[n, k] DivisorSigma[1, k] DivisorSigma[1, n - k + 1], {k, 1, n}], {n, 1, 30}]
nmax = 30; CoefficientList[Series[(1/2) D[Sum[DivisorSigma[1, k] x^k/k!, {k, 1, nmax}]^2, x], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = sum(k=1, n, binomial(n,k)*sigma(k)*sigma(n-k+1)); \\ Michel Marcus, Dec 05 2019

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

cp's OEIS Frontend

A160399 a(n) = Sum_{k=1..n} binomial(n,k) * d(k), where d(k) = the number of positive divisors of k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Formula

Extensions

A103446 Unlabeled analog of A025168.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A320568 a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*sigma(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A222115 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * sigma(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

PARI

A320568 a(n) = Sum_{k=1..n} (-1)^(n-k)binomial(n,k)sigma(k).