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 10 results.

A001787 a(n) = n*2^(n-1).

Original entry on oeis.org

0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120, 11264, 24576, 53248, 114688, 245760, 524288, 1114112, 2359296, 4980736, 10485760, 22020096, 46137344, 96468992, 201326592, 419430400, 872415232, 1811939328, 3758096384, 7784628224, 16106127360, 33285996544
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of edges in an n-dimensional hypercube.
Number of 132-avoiding permutations of [n+2] containing exactly one 123 pattern. - Emeric Deutsch, Jul 13 2001
Number of ways to place n-1 nonattacking kings on a 2 X 2(n-1) chessboard for n >= 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001
Arithmetic derivative of 2^n: a(n) = A003415(A000079(n)). - Reinhard Zumkeller, Feb 26 2002
(-1) times the determinant of matrix A_{i,j} = -|i-j|, 0 <= i,j <= n.
a(n) is the number of ones in binary numbers 1 to 111...1 (n bits). a(n) = A000337(n) - A000337(n-1) for n = 2,3,... . - Emeric Deutsch, May 24 2003
The number of 2 X n 0-1 matrices containing n+1 1's and having no zero row or column. The number of spanning trees of the complete bipartite graph K(2,n). This is the case m = 2 of K(m,n). See A072590. - W. Edwin Clark, May 27 2003
Binomial transform of 0,1,2,3,4,5,... (A001477). Without the initial 0, binomial transform of odd numbers.
With an additional leading zero, [0,0,1,4,...] this is the binomial transform of the integers repeated A004526. Its formula is then (2^n*(n-1) + 0^n)/4. - Paul Barry, May 20 2003
Number of zeros in all different (n+1)-bit integers. - Ralf Stephan, Aug 02 2003
From Lekraj Beedassy, Jun 03 2004: (Start)
Final element of a summation table (as opposed to a difference table) whose first row consists of integers 0 through n (or first n+1 nonnegative integers A001477); illustrating the case n=5:
0 1 2 3 4 5
1 3 5 7 9
4 8 12 16
12 20 28
32 48
80
and the final element is a(5)=80. (End)
This sequence and A001871 arise in counting ordered trees of height at most k where only the rightmost branch at the root actually achieves this height and the count is by the number of edges, with k = 3 for this sequence and k = 4 for A001871.
Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all elements x,y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. Then a(n) = |R|. - Ross La Haye, Sep 21 2004
Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1) and (10;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004
Number of subsequences 00 in all binary words of length n+1. Example: a(2)=4 because in 000,001,010,011,100,101,110,111 the sequence 00 occurs 4 times. - Emeric Deutsch, Apr 04 2005
If you expand the n-factor expression (a+1)*(b+1)*(c+1)*...*(z+1), there are a(n) variables in the result. For example, the 3-factor expression (a+1)*(b+1)*(c+1) expands to abc+ab+ac+bc+a+b+c+1 with a(3) = 12 variables. - David W. Wilson, May 08 2005
An inverse Chebyshev transform of n^2, where g(x)->(1/sqrt(1-4*x^2))*g(x*c(x^2)), c(x) the g.f. of A000108. - Paul Barry, May 13 2005
Sequences A018215 and A058962 interleaved. - Graeme McRae, Jul 12 2006
The number of never-decreasing positive integer sequences of length n with a maximum value of 2*n. - Ben Paul Thurston, Nov 13 2006
Total size of all the subsets of an n-element set. For example, a 2-element set has 1 subset of size 0, 2 subsets of size 1 and 1 of size 2. - Ross La Haye, Dec 30 2006
Convolution of the natural numbers [A000027] and A045623 beginning [0,1,2,5,...]. - Ross La Haye, Feb 03 2007
If M is the matrix (given by rows) [2,1;0,2] then the sequence gives the (1,2) entry in M^n. - Antonio M. Oller-Marcén, May 21 2007
If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n > 0, a(n) is equal to the number of (n+1)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007
Number of n-permutations of 3 objects u,v,w, with repetition allowed, containing exactly one u. Example: a(2)=4 because we have uv, vu, uw and wu. - Zerinvary Lajos, Dec 27 2007
A member of the family of sequences defined by a(n) = n*[c(1)*...*c(r)]^(n-1); c(i) integer. This sequence has c(1)=2, A027471 has c(1)=3. - Ctibor O. Zizka, Feb 23 2008
a(n) is the number of ways to split {1,2,...,n-1} into two (possibly empty) complementary intervals {1,2,...,i} and {i+1,i+2,...,n-1} and then select a subset from each interval. - Geoffrey Critzer, Jan 31 2009
Equals the Jacobsthal sequence A001045 convolved with A003945: (1, 3, 6, 12, ...). - Gary W. Adamson, May 23 2009
Starting with offset 1 = A059570: (1, 2, 6, 14, 34, ...) convolved with (1, 2, 2, 2, ...). - Gary W. Adamson, May 23 2009
Equals the first left hand column of A167591. - Johannes W. Meijer, Nov 12 2009
The number of tatami tilings of an n X n square with n monomers is n*2^(n-1). - Frank Ruskey, Sep 25 2010
Under T. D. Noe's variant of the hypersigma function, this sequence gives hypersigma(2^n): a(n) = A191161(A000079(n)). - Alonso del Arte, Nov 04 2011
Number of Dyck (n+2)-paths with exactly one valley at height 1 and no higher valley. - David Scambler, Nov 07 2011
Equals triangle A059260 * A016777 as a vector, where A016777 = (3n + 1): [1, 4, 7, 10, 13, ...]. - Gary W. Adamson, Mar 06 2012
Main transitions in systems of n particles with spin 1/2 (see A212697 with b=2). - Stanislav Sykora, May 25 2012
Let T(n,k) be the triangle with (first column) T(n,1) = 2*n-1 for n >= 1, otherwise T(n,k) = T(n,k-1) + T(n-1,k-1), then a(n) = T(n,n). - J. M. Bergot, Jan 17 2013
Sum of all parts of all compositions (ordered partitions) of n. The equivalent sequence for partitions is A066186. - Omar E. Pol, Aug 28 2013
Starting with a(1)=1: powers of 2 (A000079) self-convolved. - Bob Selcoe, Aug 05 2015
Coefficients of the series expansion of the normalized Schwarzian derivative -S{p(x)}/6 of the polynomial p(x) = -(x-x1)*(x-x2) with x1 + x2 = 1 (cf. A263646). - Tom Copeland, Nov 02 2015
a(n) is the number of North-East lattice paths from (0,0) to (n+1,n+1) that have exactly one east step below y = x-1 and no east steps above y = x+1. Details can be found in Pan and Remmel's link. - Ran Pan, Feb 03 2016
Also the number of maximal and maximum cliques in the n-hypercube graph for n > 0. - Eric W. Weisstein, Dec 01 2017
Let [n]={1,2,...,n}; then a(n-1) is the total number of elements missing in proper subsets of [n] that contain n to form [n]. For example, for n = 3, a(2) = 4 since the proper subsets of [3] that contain 3 are {3}, {1,3}, {2,3} and the total number of elements missing in these subsets to form [3] is 4: 2 in the first subset, 1 in the second, and 1 in the third. - Enrique Navarrete, Aug 08 2020
Number of 3-permutations of n elements avoiding the patterns 132, 231. See Bonichon and Sun. - Michel Marcus, Aug 19 2022

Examples

			a(2)=4 since 2314, 2341,3124 and 4123 are the only 132-avoiding permutations of 1234 containing exactly one increasing subsequence of length 3.
x + 4*x^2 + 12*x^3 + 32*x^4 + 80*x^5 + 192*x^6 + 448*x^7 + ...
a(5) = 1*0 + 5*1 + 10*2 + 10*3 + 5*4 + 1*5 = 80, with 1,5,10,10,5,1 the 5th row of Pascal's triangle. - _J. M. Bergot_, Apr 29 2014

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 131.
  • Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 282.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Three other versions, essentially identical, are A085750, A097067, A118442.
Partial sums of A001792.
A058922(n+1) = 4*A001787(n).
Equals A090802(n, 1).
Column k=1 of A038207.
Row sums of A003506, A322427, A322428.
Cf. A053109, A001788, A001789, A000337, A130300, A134083, A002064, A027471, A003945, A059670, A167591, A059260, A016777, A212697, A000079, A263646.

Programs

  • Haskell
    a001787 n = n * 2 ^ (n - 1)
a001787_list = zipWith (*) [0..] $ 0 : a000079_list
-- Reinhard Zumkeller, Jul 11 2014
  • Magma
    [n*2^(n-1): n in [0..40]]; // Vincenzo Librandi, Feb 04 2016
  • Maple
    spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, B, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..29); # Zerinvary Lajos, Oct 09 2006
A001787:=1/(2*z-1)^2; # Simon Plouffe in his 1992 dissertation, dropping the initial zero
  • Mathematica
    Table[Sum[Binomial[n, i] i, {i, 0, n}], {n, 0, 30}] (* Geoffrey Critzer, Mar 18 2009 *)
f[n_] := n 2^(n - 1); f[Range[0, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
Array[# 2^(# - 1) &, 40, 0] (* Harvey P. Dale, Jul 26 2011 *)
Join[{0}, Table[n 2^(n - 1), {n, 20}]] (* Eric W. Weisstein, Dec 01 2017 *)
Join[{0}, LinearRecurrence[{4, -4}, {1, 4}, 20]] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[x/(-1 + 2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)
  • PARI
    {a(n) = if( n<0, 0, n * 2^(n-1))}
  • PARI
    concat(0, Vec(x/(1-2*x)^2 + O(x^50))) \\ Altug Alkan, Nov 03 2015
  • Python
    def A001787(n): return n*(1<Chai Wah Wu, Nov 14 2022

Formula

a(n) = Sum_{k=1..n} k*binomial(n, k). - Benoit Cloitre, Dec 06 2002
E.g.f.: x*exp(2x). - Paul Barry, Apr 10 2003
G.f.: x/(1-2*x)^2.
G.f.: x / (1 - 4*x / (1 + x / (1 - x))). - Michael Somos, Apr 07 2012
A108666(n) = Sum_{k=0..n} binomial(n, k)^2 * a(n). - Michael Somos, Apr 07 2012
PSumSIGN transform of A053220. PSumSIGN transform is A045883. Binomial transform is A027471(n+1). - Michael Somos, Jul 10 2003
Starting at a(1)=1, INVERT transform is A002450, INVERT transform of A049072, MOBIUS transform of A083413, PSUM transform is A000337, BINOMIAL transform is A081038, BINOMIAL transform of A005408. - Michael Somos, Apr 07 2012
a(n) = 2*a(n-1)+2^(n-1).
a(2*n) = n*4^n, a(2*n+1) = (2*n+1)4^n.
G.f.: x/det(I-x*M) where M=[1,i;i,1], i=sqrt(-1). - Paul Barry, Apr 27 2005
Starting 1, 1, 4, 12, ... this is 0^n + n2^(n-1), the binomial transform of the 'pair-reversed' natural numbers A004442. - Paul Barry, Jul 24 2003
Convolution of [1, 2, 4, 8, ...] with itself. - Jon Perry, Aug 07 2003
The signed version of this sequence, n(-2)^(n-1), is the inverse binomial transform of n(-1)^(n-1) (alternating sign natural numbers). - Paul Barry, Aug 20 2003
a(n-1) = (Sum_{k=0..n} 2^(n-k-1)*C(n-k, k)*C(1,(k+1)/2)*(1-(-1)^k)/2) - 0^n/4. - Paul Barry, Oct 15 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)(n-2k)^2. - Paul Barry, May 13 2005
a(n+2) = A049611(n+2) - A001788(n).
a(n) = n! * Sum_{k=0..n} 1/((k - 1)!(n - k)!). - Paul Barry, Mar 26 2003
a(n+1) = Sum_{k=0..n} 4^k * A109466(n,k). - Philippe Deléham, Nov 13 2006
Row sums of A130300 starting (1, 4, 12, 32, ...). - Gary W. Adamson, May 20 2007
Equals row sums of triangle A134083. Equals A002064(n) + (2^n - 1). - Gary W. Adamson, Oct 07 2007
a(n) = 4*a(n-1) - 4*a(n-2), a(0)=0, a(1)=1. - Philippe Deléham, Nov 16 2008
Sum_{n>0} 1/a(n) = 2*log(2). - Jaume Oliver Lafont, Feb 10 2009
a(n) = A000788(A000225(n)) = A173921(A000225(n)). - Reinhard Zumkeller, Mar 04 2010
a(n) = n * A011782(n). - Omar E. Pol, Aug 28 2013
a(n-1) = Sum_{t_1+2*t_2+...+n*t_n=n} (t_1+t_2+...+t_n-1)*multinomial(t_1+t_2 +...+t_n,t_1,t_2,...,t_n). - Mircea Merca, Dec 06 2013
a(n+1) = Sum_{r=0..n} (2*r+1)*C(n,r). - J. M. Bergot, Apr 07 2014
a(n) = A007283(n)*n/6. - Enxhell Luzhnica, Apr 16 2016
a(n) = (A000225(n) + A000337(n))/2. - Anton Zakharov, Sep 17 2016
Sum_{n>0} (-1)^(n+1)/a(n) = 2*log(3/2) = 2*A016578. - Ilya Gutkovskiy, Sep 17 2016
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (i+1) * C(k,i). - Wesley Ivan Hurt, Sep 21 2017
a(n) = Sum_{i=1..n} Sum_{j=1..n} phi(i)*binomial(n, i*j). - Ridouane Oudra, Feb 17 2024

A034691 Euler transform of powers of 2 [1,2,4,8,16,...].

Original entry on oeis.org

1, 1, 3, 7, 18, 42, 104, 244, 585, 1373, 3233, 7533, 17547, 40591, 93711, 215379, 493735, 1127979, 2570519, 5841443, 13243599, 29953851, 67604035, 152258271, 342253980, 767895424, 1719854346, 3845443858
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This is the number of different hierarchical orderings that can be formed from n unlabeled elements: these are divided into groups and the elements in each group are then arranged in an "unlabeled preferential arrangement" or "composition" as in A000079. - Thomas Wieder and N. J. A. Sloane, Jun 10 2003
From Gus Wiseman, Mar 03 2016: (Start)
The original Sloane-Wieder definition, "To obtain a hierarchical ordering we partition the elements into unlabeled nonempty subsets and form a composition of each subset," [arXiv:math/0307064] has two possible meanings. The first possible meaning is that we should (1) choose a set partition pi of {1...n} and (2) for each block of pi choose a composition of the number of elements. In this case the correct number of such structures would evidently be counted by A004211 which differs from a(n) for n > 2.
The other possible meaning is that after we have done (1) and (2) above we (3) "forget" the choice of pi. We will have produced a collection M of multisets of compositions. The span of M (its set of distinct elements) is correctly counted by A034691 and it seems that non-isomorphic hierarchical orderings of unlabeled sets are nothing more than multisets of compositions. This discovery is due to Wieder. (End)
The asymptotic formula in the article by N. J. A. Sloane and Thomas Wieder, "The Number of Hierarchical Orderings" (Theorem 3) is incorrect (a multiplicative factor of 1.397... is missing, see my formula below). - Vaclav Kotesovec, Sep 08 2014
Number of partitions of n into 1 sort of 1, 2 sorts of 2's, 4 sorts of 3's, ..., 2^(k-1) sorts of k's, ... . - Joerg Arndt, Sep 09 2014
Also number of normal multiset partitions of weight n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Mar 03 2016

Examples

			The normal multiset partitions for a(4) = 18: {{1111},{1222},{1122},{1112},{1233},{1223},{1123},{1234},{1,111},{1,122},{1,112},{1,123},{11,11},{11,12},{12,12},{1,1,11},{1,1,12},{1,1,1,1}}

Links

Crossrefs

Cf. A034899, A075729, A247003, A004211, A104500 (Euler transform), A290222 (Multiset transform).

Programs

  • Maple
    oo := 101: mul( 1/(1-x^j)^(2^(j-1)),j=1..oo): series(%,x,oo): t1 := seriestolist(%); A034691 := n-> t1[n+1];
with(combstruct); SetSeqSetU := [T, {T=Set(S), S=Sequence(U,card >= 1), U=Set(Z,card >=1)},unlabeled]; seq(count(SetSeqSetU,size=j),j=1..12);
# Alternative, uses EulerTransform from A358369:
a := EulerTransform(BinaryRecurrenceSequence(2, 0)):
seq(a(n), n = 0..27); # Peter Luschny, Nov 17 2022
  • Mathematica
    nn = 30; b = Table[2^n, {n, 0, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}],  x] (* T. D. Noe, Nov 21 2011 *)
Table[SeriesCoefficient[E^(Sum[x^k/(1 - 2*x^k)/k, {k, 1, n}]), {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 08 2014 *)
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
allnmsp[0]={};allnmsp[1]={{{1}}};allnmsp[n_Integer]:=allnmsp[n]=Join[allnmsp[n-1],List/@allnorm[n],Join@@Function[ptn,Append[ptn,#]&/@Select[allnorm[n-Length[Join@@ptn]],OrderedQ[{Last[ptn],#}]&]]/@allnmsp[n-1]];
Apply[SequenceForm,Select[allnmsp[4],Length[Join@@#]===4&],{2}] (* to construct the example *)
Table[Length[Complement[allnmsp[n],allnmsp[n-1]]],{n,1,8}] (* Gus Wiseman, Mar 03 2016 *)
  • PARI
    A034691(n,l=1+O('x^(n+1)))={polcoeff(1/prod(k=1,n,(l-'x^k)^2^(k-1)),n)} \\ Michael Somos, Nov 21 2011, edited by M. F. Hasler, Jul 24 2017
  • SageMath
    # uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(2, 0)
b = EulerTransform(a)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

Formula

G.f.: 1 / Product_{n>=1} (1-x^n)^(2^(n-1)).
Recurrence: a(n) = (1/n) * Sum_{m=1..n} a(n-m)*c(m) where c(m) = A083413(m).
a(n) ~ c * 2^n * exp(sqrt(2*n)) / (sqrt(2*Pi) * exp(1/4) * 2^(3/4) * n^(3/4)), where c = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502... (see A247003). - Vaclav Kotesovec, Sep 08 2014

A179470 G.f. satisfies A(x) = exp( Sum_{n>=1} A(2*x^n)*x^n/n ).

Original entry on oeis.org

1, 1, 3, 15, 138, 2370, 78532, 5110472, 659436845, 169486506217, 86947958127377, 89122003350193045, 182611160539104099261, 748158103862060509908713, 6129659711065116858192667033, 100434475863953990317790200253757
Offset: 0

Views

Author

Paul D. Hanna, Jul 15 2010

Keywords

Comments

Compare to the g.f. of A000081: G(x) = exp( Sum_{n>=1} G(x^n)*x^n/n ).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 138*x^4 + 2370*x^5 +...
log(A(x)) = A(2x)*x + A(2x^2)*x^2/2 + A(2x^3)*x^3/3 + A(2x^4)*x^4/4 + A(2x^5)*x^5/5 +...
More generally, if F(x,q) = exp( Sum_{n>=1} F(q*x^n,q)*x^n/n )
then coefficients in F(x,q) = Sum_{n>=0} c(n,q)*x^n begin:
c(0,q) = 1; c(1,q) = 1; c(2,q) = q + 1;
c(3,q) = q^3 + q^2 + q + 1;
c(4,q) = q^6 + q^5 + q^4 + 2*q^3 + 3/2*q^2 + 3/2*q + 1;
c(5,q) = q^10 + q^9 + q^8 + 2*q^7 + 5/2*q^6 + 5/2*q^5 + 3*q^4 + 3*q^3 + 3/2*q^2 + 3/2*q + 1; ...
where C(n,q) are integers for integer values of q.

Links

Crossrefs

Cf. A000081, A083413.

Programs

  • PARI
    {a(n)=my(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A,x,2*x^m+x*O(x^n))*x^m/m)));polcoeff(A,n)}

Formula

From Seiichi Manyama, Jun 01 2023: (Start)
A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(k+1))^(2^k * a(k)).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * 2^(d-1) * a(d-1) ) * a(n-k). (End)

A090879 a(n) = Sum_{d|n} d*2^(n-d).

Original entry on oeis.org

1, 4, 7, 20, 21, 94, 71, 328, 457, 1194, 1035, 7052, 4109, 17294, 33807, 83984, 65553, 389650, 262163, 1484820, 1949717, 4216854, 4194327, 29409304, 22020121, 67215386, 119799835, 350453788, 268435485, 1755807774, 1073741855
Offset: 1

Views

Author

Vladeta Jovovic, Feb 14 2004

Keywords

Links

Crossrefs

Cf. A074854, A083413.

Programs

  • Mathematica
    f[n_] := Block[{d = Divisors[n]}, Apply[Plus, d*2^(n - d)]]; Table[ f[n], {n, 1, 32}] (* Robert G. Wilson v, Feb 16 2004 *)

Formula

G.f.: Sum_{m>0} m*x^m/(1-(2*x)^m).

Extensions

More terms from Robert G. Wilson v, Feb 16 2004

A359018 a(n) = Sum_{d|n} d * 3^(d-1).

Original entry on oeis.org

1, 7, 28, 115, 406, 1492, 5104, 17611, 59077, 197242, 649540, 2127364, 6908734, 22325632, 71744968, 229600123, 731794258, 2324583475, 7360989292, 23245426690, 73222477552, 230128420012, 721764371008, 2259438436708, 7060738412431, 22029510754258, 68630377423960
Offset: 1

Views

Author

Seiichi Manyama, Dec 19 2022

Keywords

Links

Crossrefs

Cf. A002129, A034730, A083413, A167531, A359186, A359189.

Programs

  • Magma
    A359018:= func< n | (&+[3^(d-1)*d: d in Divisors(n)]) >;
[A359018(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
  • Mathematica
    a[n_] := DivisorSum[n, 3^(#-1)*# &]; Array[a, 27] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d*3^(d-1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-3*x^k)^2))
  • SageMath
    def A359018(n): return sum(3^(k-1)*k for k in (1..n) if (k).divides(n))
[A359018(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

Formula

G.f.: Sum_{k>=1} x^k/(1 - 3 * x^k)^2.

A200751 Expansion of Product_{k>0} (1 - x^k)^(2^(k-1)) in powers of x.

Original entry on oeis.org

1, -1, -2, -2, -3, -1, -2, 6, 12, 36, 74, 162, 301, 599, 1090, 1986, 3479, 5993, 9852, 15644, 23094, 30690, 31868, 9068, -82372, -345308, -1010956, -2577868, -6098822, -13751218, -29962588, -63604140, -132205949, -269982371, -542866266, -1076420666
Offset: 0

Views

Author

Michael Somos, Nov 21 2011

Keywords

Examples

			1 - x - 2*x^2 - 2*x^3 - 3*x^4 - x^5 - 2*x^6 + 6*x^7 + 12*x^8 + 36*x^9 + ...

Links

Crossrefs

Cf. A034691, A083413, A360634.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Series[Product[(1 - x^k)^2^(k - 1),
{k, n}], {x, 0, n}], n]; Table[a[n], {n, 0, 35}] (* T. D. Noe, Nov 23 2011 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod( k=1, n, (1 - x^k + A) ^ 2^(k - 1)), n))}

Formula

Let F(a, x) = (1 - a) * (1 - a*x)^2 * (1 - a*x^2)^4 * ... where |x|<1/2. Then F(a, x) = (1 - a) * F(a*x, x)^2 and g.f. A(x) = F(x, x).
Euler transform of [ -1, -2, -4, -8, -16, ... ].
G.f.: (1 - x) * (1 - x^2)^2 * (1 - x^3)^4 * ...
Convolution inverse of A034691.
a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A083413(k) * a(n-k). - Seiichi Manyama, Jul 17 2023
a(n) = Sum_{k=0..2n} (-1)^k*A360634(2n,k). - Alois P. Heinz, Sep 09 2023

A359186 a(n) = Sum_{d|n} d * 4^(d-1).

Original entry on oeis.org

1, 9, 49, 265, 1281, 6201, 28673, 131337, 589873, 2622729, 11534337, 50338105, 218103809, 939552777, 4026533169, 17180000521, 73014444033, 309238241337, 1305670057985, 5497560761865, 23089744212017, 96757034778633, 404620279021569, 1688849910733113
Offset: 1

Views

Author

Seiichi Manyama, Dec 19 2022

Keywords

Crossrefs

Cf. A002129, A083413, A359018.
Cf. A167531, A339684, A359190.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 4^(#-1)*# &]; Array[a, 24] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d*4^(d-1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-4*x^k)^2))

Formula

G.f.: Sum_{k>=1} x^k/(1 - 4 * x^k)^2.

A074225 a(n) = n * Sum_{d|n} d*2^(d-1).

Original entry on oeis.org

1, 10, 39, 148, 405, 1254, 3143, 8488, 20853, 52050, 123915, 297804, 692237, 1611974, 3687795, 8405584, 18939921, 42512562, 94634003, 209819940, 462431697, 1015269486, 2218786839, 4832458392, 10485762025, 22684180610, 48922424415, 105229923596
Offset: 1

Views

Author

N. J. A. Sloane, Jun 15 2003

Keywords

Links

Crossrefs

Equals n*A083413(n).

Programs

  • Mathematica
    a[n_] := n*DivisorSum[n, #*2^(#-1)&]; Array[a, 30] (* Jean-François Alcover, Dec 03 2015 *)
  • PARI
    vector(100, n, n*sumdiv(n, d, d*2^(d-1))) \\ Colin Barker, Jan 29 2015

A077272 a(n) = Sum_{d|n} d^2*2^(d-1)*(n/d-1) for n > 0.

Original entry on oeis.org

0, 0, 1, 2, 11, 4, 57, 6, 159, 80, 441, 10, 1567, 12, 3197, 958, 8647, 16, 23301, 18, 53003, 6508, 124005, 22, 315755, 1624, 692345, 41786, 1615939, 28, 3795873, 30, 8414231, 248200, 18940065, 14978, 43126711, 36, 94634165, 1384918, 209905711, 40, 465656541
Offset: 0

Views

Author

N. J. A. Sloane, Jun 13 2003

Keywords

Examples

			a(4) = 1*1*3 + 4*2*1 + 16*8*0 = 11.

Crossrefs

Cf. A083413.

Formula

G.f.: Sum_{m>0} (m-1)*x^m*(1+2*x^m)/(1-2*x^m)^3. - Vladeta Jovovic, Jun 21 2003

A318368 a(n) = Sum_{d|n} (-1)^(n/d+1)*d*2^(d-1).

Original entry on oeis.org

1, 3, 13, 27, 81, 183, 449, 987, 2317, 5043, 11265, 24399, 53249, 114243, 245853, 523227, 1114113, 2357175, 4980737, 10480587, 22020557, 46126083, 96468993, 201302799, 419430481, 872361987, 1811941645, 3757981275, 7784628225, 16105886823, 33285996545, 68718951387, 141733932045
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 24 2018

Keywords

Crossrefs

Cf. A001787, A054599, A083413, A098407.

Programs

  • Mathematica
    Table[Sum[(-1)^(n/d + 1) d 2^(d - 1), {d, Divisors[n]}], {n, 33}]
nmax = 33; Rest[CoefficientList[Series[Sum[k 2^(k - 1) x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 33; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(2^(k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d+1)*d*2^(d-1)); \\ Michel Marcus, Aug 25 2018

Formula

G.f.: Sum_{k>=1} k*2^(k-1)*x^k/(1 + x^k).
L.g.f.: log(Product_{k>=1} (1 + x^k)^(2^(k-1))) = Sum_{n>=1} a(n)*x^n/n.
a(n) ~ n * 2^(n-1). - Vaclav Kotesovec, Aug 25 2018
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