A036799 -id:A036799 - cp's OEIS frontend

A036289 a(n) = n*2^n.

0, 2, 8, 24, 64, 160, 384, 896, 2048, 4608, 10240, 22528, 49152, 106496, 229376, 491520, 1048576, 2228224, 4718592, 9961472, 20971520, 44040192, 92274688, 192937984, 402653184, 838860800, 1744830464, 3623878656, 7516192768, 15569256448, 32212254720

Offset: 0

N. J. A. Sloane, Dec 11 1999

Right side of the binomial sum Sum_{i = 0..n} (n-2*i)^2 * binomial(n, i) = n*2^n. - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
Let W 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, or y is a proper subset of x and there are no z in P(A) such that y is a proper subset of z and z is a proper subset of x. Then a(n) = |W|. - Ross La Haye, Sep 26 2007
Partial sums give A036799. - Vladimir Joseph Stephan Orlovsky, Jul 09 2011
a(n) = n with the bits shifted to the left by n places (new bits on the right hand side are zeros). - Indranil Ghosh, Jan 05 2017
Satisfies Benford's law [Theodore P. Hill, Personal communication, Feb 06, 2017]. - N. J. A. Sloane, Feb 08 2017
Also the circumference of the n-cube connected cycle graph. - Eric W. Weisstein, Sep 03 2017
a(n) is also the number of derangements in S_{n+3} with a descent set of {i, i+1} such that i ranges from 1 to n-2. - Isabella Huang, Mar 17 2018
a(n-1) is also the number of multiplications required to compute the permanent of general n X n matrices using Glynn's formula (see Theorem 2.1 in Glynn). - Stefano Spezia, Oct 27 2021

Arno Berger and Theodore P. Hill. An Introduction to Benford's Law. Princeton University Press, 2015.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.29)

Indranil Ghosh, Table of n, a(n) for n = 0..1000 (first 501 terms from T. D. Noe)
C. Banderier and S. Schwer, Why Delannoy numbers?, arXiv:math/0411128 [math.CO], 2004.
David G. Glynn, The permanent of a square matrix, European Journal of Combinatorics, Volume 31, Issue 7, 2010, pp. 1887-1891.
A. F. Horadam, Oresme numbers, Fib. Quart., 12 (1974), 267-271.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Cube-Connected Cycle Graph.
Eric Weisstein's World of Mathematics, Graph Circumference.
Index entries for linear recurrences with constant coefficients, signature (4,-4).
Index entries for sequences related to Benford's law

Equals 2*A001787. Equals A003261(n) + 1.

Cf. A000079, A036799, A096195, A097064.

a036289 n = n * 2 ^ n
a036289_list = zipWith (*) [0..] a000079_list
-- Reinhard Zumkeller, Mar 05 2012

g:=1/(1-2*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)*n, n=0..34); # Zerinvary Lajos, Jan 11 2009

Table[n*2^n, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2010 *)
LinearRecurrence[{4,-4},{0,2},40] (* Harvey P. Dale, Mar 02 2018 *)

a(n)=n<Charles R Greathouse IV, Jun 15 2011

a=lambda n: n<Indranil Ghosh, Jan 05 2017

Main diagonal of array (A085454) defined by T(i, 1) = i, T(1, j) = 2j, T(i, j) = T(i-1, j) + T(i-1, j-1). - Benoit Cloitre, Aug 05 2003
Binomial transform of A005843, the even numbers. - Joshua Zucker, Jan 13 2006
G.f.: 2*x/(1-2*x)^2. - R. J. Mathar, Nov 21 2007
a(n) = A000079(n)*n. - Omar E. Pol, Dec 21 2008
E.g.f.: 2*x exp(2*x). - Geoffrey Critzer, Oct 03 2011
a(n) = A002064(n) - 1. - Reinhard Zumkeller, Mar 16 2013
From Vaclav Kotesovec, Feb 14 2015: (Start)
Sum_{n>=1} 1/a(n) = log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(3/2).
(End)

A000337 a(n) = (n-1)*2^n + 1.

Original entry on oeis.org

0, 1, 5, 17, 49, 129, 321, 769, 1793, 4097, 9217, 20481, 45057, 98305, 212993, 458753, 983041, 2097153, 4456449, 9437185, 19922945, 41943041, 88080385, 184549377, 385875969, 805306369, 1677721601, 3489660929, 7247757313, 15032385537, 31138512897, 64424509441

Offset: 0

N. J. A. Sloane

a(n) also gives number of 0's in binary numbers 1 to 111..1 (n+1 bits). - Stephen G Penrice, Oct 01 2000
Numerator of m(n) = (m(n-1)+n)/2, m(0)=0. Denominator is A000079. - Reinhard Zumkeller, Feb 23 2002
a(n) is the number of directed column-convex polyominoes of area n+2 having along the lower contour exactly one vertical step that is followed by a horizontal step (a reentrant corner). - Emeric Deutsch, May 21 2003
a(n) is the number of bits in binary numbers from 1 to 111...1 (n bits). Partial sums of A001787. - Emeric Deutsch, May 24 2003
Genus of graph of n-cube = a(n-3) = 1+(n-4)*2^(n-3), n>1.
Sum of ordered partitions of n where each element is summed via T(e-1). See A066185 for more information. - Jon Perry, Dec 12 2003
a(n-2) is the number of Dyck n-paths with exactly one peak at height >= 3. For example, there are 5 such paths with n=4: UUUUDDDD, UUDUUDDD, UUUDDUDD, UDUUUDDD, UUUDDDUD. - David Callan, Mar 23 2004
Permutations in S_{n+2} avoiding 12-3 that contain the pattern 13-2 exactly once.
a(n) is prime for n = 2, 3, 7, 27, 51, 55, 81. a(n) is semiprime for n = 4, 5, 6, 8, 9, 10, 11, 13, 15, 19, 28, 32, 39, 57, 63, 66, 75, 97. - Jonathan Vos Post, Jul 18 2005
A member of the family of sequences defined by a(n) = Sum_{i=1..n} i*[c(1)*...*c(r)]^(i-1). This sequence has c(1)=2, A014915 has c(1)=3. - Ctibor O. Zizka, Feb 23 2008
Starting with 1 = row sums of A023758 as a triangle by rows: [1; 2,3; 4,6,7; 8,12,14,15; ...]. - Gary W. Adamson, Jul 18 2008
Equivalent formula given in Brehm: for each q >= 3 there exists a polyhedral map M_q of type {4, q} with [number of vertices] f_0 = 2^q and [genus] g = (2^(q-3))*(q-4) + 1 such that M_q and its dual have polyhedral embeddings in R^3 [McMullen et al.]. - Jonathan Vos Post, Jul 25 2009
Sums of rows of the triangle in A173787. - Reinhard Zumkeller, Feb 28 2010
This sequence is related to A000079 by a(n) = n*A000079(n)-Sum_{i=0..n-1} A000079(i). - Bruno Berselli, Mar 06 2012
(1 + 5*x + 17*x^2 + 49*x^3 + ...) = (1 + 2*x + 4*x^2 + 8*x^3 + ...) * (1 + 3*x + 7*x^2 + 15*x^3 + ...). - Gary W. Adamson, Mar 14 2012
The first barycentric coordinate of the centroid of Pascal triangles, assuming that numbers are weights, is A000295(n+1)/A000337(n), no matter what the triangle sides are. See attached figure. - César Eliud Lozada, Nov 14 2014
a(n) is the n-th number that is a sum of n positive n-th powers for n >= 1. a(4) = 49 = A003338(4). - Alois P. Heinz, Aug 01 2020
a(n) is the sum of the largest elements of all subsets of {1,2,..,n}. For example, a(3)=17; the subsets of {1,2,3} are {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, and the sum of the largest elements is 17. - Enrique Navarrete, Aug 20 2020
a(n-1) is the sum of the second largest elements of the subsets of {1,2,..,n} that contain n. For example, for n = 4, a(3)=17; the subsets of {1,2,3,4} that contain 4 are {4}, {1,4}, {2,4}, {3,4}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}, and the sum of the second largest elements is 17. - Enrique Navarrete, Aug 24 2020
a(n-1) is also the sum of diameters of all subsets of {1,2,...,n} that contain n.  For example, for n = 4, a(3)=17; the subsets of {1,2,3,4} that contain 4 are {4}, {1,4}, {2,4}, {3,4}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}; the diameters of these sets are 0,3,2,1,3,3,2,3 and the sum is 17. - Enrique Navarrete, Sep 07 2020
a(n-1) is also the number of additions required to compute the permanent of general n X n matrices using trellis methods (see Theorems 5 and 6, pp. 10-11 in Kiah et al.). - Stefano Spezia, Nov 02 2021

F. Harary, Topological concepts in graph theory, pp. 13-17 of F. Harary and L. Beineke, editors, A seminar on Graph Theory, Holt, Rinehart and Winston, New York, 1967.
V. G. Gutierrez and S. L. de Medrano, Surfaces as complete intersections, in Riemann and Klein Surfaces, Automorphisms, Symmetries and Moduli Spaces, edited by Milagros Izquierdo, S. Allen Broughton, Antonio F. Costa, Contemp. Math. vol. 629, 2014, pp. 171-.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 119.
G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers of G. H. Hardy, Vol. VII, p. 430.
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).

Indranil Ghosh, Table of n, a(n) for n = 0..1000 (first 301 terms from T. D. Noe)
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 14.
H. H. Bauschke and R. M. Corless, Analyzing a Projection Method with Maple, MapleTech Journal, 4:1 (1997), 2-7.
L. W. Beineke and F. Harary, The genus of the n-cube, Canad. J. Math., 17 (1965), 494-496.
Ulrich Brehm and Egon Schulte, Polyhedral Maps. [_Jonathan Vos Post_, Jul 25 2009]
Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
Pedro Fernando Fernández Espinosa and Agustín Moreno Cañadas, Homological Ideals as Integer Specializations of Some Brauer Configuration Algebras, arXiv:2104.00050 [math.RT], 2021.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
Han Mao Kiah, Alexander Vardy, and Hanwen Yao, Computing Permanents on a Trellis, arXiv:2107.07377 [cs.IT], 2021.
S. Kitaev, J. Remmel, and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012.
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 2015, #A16. (arXiv, arXiv:1302.2274 [math.CO], 2013)
Santiago López de Medrano, On the genera of moment-angle manifolds associated to dual-neighborly polytopes, combinatorial formulas and sequences, arXiv:2003.07508 [math.GT], 2020.
César Eliud Lozada, Centroids of Pascal triangles
P. McMullen, Ch. Schulz and J.M. Wills, Polyhedral manifolds in E^3 with unusually large genus, Israel J. Math. 46:127-144, 1983. [From _Jonathan Vos Post_, Jul 25 2009]
Toufik Mansour, Restricted permutations by patterns of type (2,1), arXiv:math/0202219 [math.CO], 2002.
Michael Penn, An awesome number theory contest problem, YouTube video, 2022.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Len Smiley, Hardy's algorithm.
Nsibiet E. Udo, Praise Adeyemo, Balazs Szendroi, and Stavros Argyrios Papadakis, Ideals, representations and a symmetrised Bernoulli triangle, arXiv:2409.10278 [math.AC], 2024. See pp. 2,4,8.
Eric Weisstein's World of Mathematics, Graph Genus.
Eric Weisstein's World of Mathematics, Hypercube Graph.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

a(n) = T(3, n), array T given by A048472. A036799/2.
Cf. A001787, A066185, A077436, A023758, A014915.
Cf. A000079, A209359.
Cf. A000295, A008292.
Cf. A003338.
Main diagonal of A336725.

List([0..30],n->(n-1)*2^n+1); # Muniru A Asiru, Oct 24 2018

[(n-1)*2^n + 1: n in [0..40]]; // Vincenzo Librandi, Nov 21 2014

A000337 := proc(n) 1+(n-1)*2^n ; end proc: # R. J. Mathar, Oct 10 2011

Table[Sum[(-1)^(n - k) k (-1)^(n - k) Binomial[n + 1, k + 1], {k, 0, n}], {n, 0, 28}] (* Zerinvary Lajos, Jul 08 2009 *)
Table[(n - 1) 2^n + 1, {n, 0, 40}] (* Harvey P. Dale, Jun 21 2011 *)
LinearRecurrence[{5, -8, 4}, {0, 1, 5}, 40] (* Harvey P. Dale, Jun 21 2011 *)
CoefficientList[Series[x / ((1 - x) (1 - 2 x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 21 2014 *)

PARI
```
a(n)=if(n<0,0,(n-1)*2^n+1)
```

a=lambda n:((n-1)<<(n))+1 # Indranil Ghosh, Jan 05 2017

Binomial transform of A004273. Binomial transform of A008574 if the leading zero is dropped.
G.f.: x/((1-x)*(1-2*x)^2). - Simon Plouffe in his 1992 dissertation
E.g.f.: exp(x) - exp(2*x)*(1-2*x). a(n) = 4*a(n-1) - 4*a(n-2)+1, n>0. Series reversion of g.f. A(x) is x*A034015(-x). - Michael Somos
Binomial transform of n/(n+1) is a(n)/(n+1). - Paul Barry, Aug 19 2005
a(n) = A119258(n+1,n-1) for n>0. - Reinhard Zumkeller, May 11 2006
Convolution of "Number of fixed points in all 231-avoiding involutions in S_n" (A059570) with "The odd numbers" (A005408), treating the result as if offset=0. - Graeme McRae, Jul 12 2006
a(n) = Sum_{k=1..n} k*2^(k-1), partial sums of A001787. - Zerinvary Lajos, Oct 19 2006
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3), n > 2. - Harvey P. Dale, Jun 21 2011
a(n) = Sum_{k=1..n} Sum_{i=1..n} i * C(k,i). - Wesley Ivan Hurt, Sep 19 2017
a(n) = A000295(n+1)^2 - A000295(n)*A000295(n+2). - Gregory Gerard Wojnar, Oct 23 2018

A130472 A permutation of the integers: a(n) = (-1)^n * floor( (n+1)/2 ).

Original entry on oeis.org

0, -1, 1, -2, 2, -3, 3, -4, 4, -5, 5, -6, 6, -7, 7, -8, 8, -9, 9, -10, 10, -11, 11, -12, 12, -13, 13, -14, 14, -15, 15, -16, 16, -17, 17, -18, 18, -19, 19, -20, 20, -21, 21, -22, 22, -23, 23, -24, 24, -25, 25, -26, 26, -27, 27, -28, 28, -29, 29, -30, 30, -31, 31, -32, 32

Offset: 0

Clark Kimberling, May 28 2007

Pisano period lengths: 1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, ... - R. J. Mathar, Aug 10 2012

Partial sums of A038608. - Stanislav Sykora, Nov 27 2013

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (-1,1,1).
Index entries for sequences that are permutations of the integers

Sums of the form Sum_{k=0..n} k^p * q^k: A059841 (p=0,q=-1), this sequence (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

Cf. A001057, A038608.

[((-1)^n*(2*n+1)-1)/4: n in [0..80]]; // Vincenzo Librandi, Mar 29 2015

A130472:=n->ceil(n/2)*(-1)^n; seq(A130472(k), k=0..50); # Wesley Ivan Hurt, Oct 22 2013

CoefficientList[Series[(x^2-x)/(1-x^2)^2, {x, 0, 64}], x] (* Geoffrey Critzer, Sep 29 2013 *)
LinearRecurrence[{-1,1,1},{0,-1,1},70] (* Harvey P. Dale, Mar 02 2018 *)

a(n)=(-1)^n*n\2 \\ Charles R Greathouse IV, Sep 02 2015

[((-1)^n*(2*n+1) - 1)/4 for n in (0..70)] # G. C. Greubel, Mar 31 2021

a(n) = -A001057(n).
a(2n) = n, a(2n+1) = -(n+1).
a(n) = Sum_{k=0..n} k*(-1)^k.
a(n) = -a(n-1) +a(n-2) +a(n-3).
G.f.: -x/( (1-x)*(1+x)^2 ). - R. J. Mathar, Feb 20 2011
a(n) = floor( (n/2)*(-1)^n ). - Wesley Ivan Hurt, Jun 14 2013
a(n) = ceiling( n/2 )*(-1)^n. - Wesley Ivan Hurt, Oct 22 2013
a(n) = ((-1)^n*(2*n+1) - 1)/4. - Adriano Caroli, Mar 28 2015
E.g.f.: (1/4)*(-exp(x) + (1-2*x)*exp(-x) ). - G. C. Greubel, Mar 31 2021

A187059 The exponent of highest power of 2 dividing the product of the elements of the n-th row of Pascal's triangle (A001142).

Original entry on oeis.org

0, 0, 1, 0, 5, 2, 4, 0, 17, 10, 12, 4, 18, 8, 11, 0, 49, 34, 36, 20, 42, 24, 27, 8, 58, 36, 39, 16, 47, 22, 26, 0, 129, 98, 100, 68, 106, 72, 75, 40, 122, 84, 87, 48, 95, 54, 58, 16, 162, 116, 119, 72, 127, 78, 82, 32, 147, 94, 98, 44, 108, 52, 57, 0, 321, 258, 260, 196, 266, 200, 203, 136, 282, 212, 215, 144, 223, 150, 154, 80, 322, 244, 247, 168, 255, 174, 178, 96, 275, 190, 194, 108, 204, 116, 121, 32, 418, 324, 327, 232, 335

Offset: 0

Bruce Reznick, Mar 05 2011

The exponent of the highest power of 2 which divides Product_{k=0..n} binomial(n, k). This can be computed using de Polignac's formula.

This is the function ord_2(Ḡ_n) extensively studied in Lagarias-Mehta (2014), and plotted in Fig. 1.1. - Antti Karttunen, Oct 22 2014

			For example, if n = 4, the power of 2 that divides 1*4*6*4*1 is 5.

I. Niven, H. S. Zuckerman, H. L. Montgomery, An Introduction to the Theory of Numbers, Wiley, 1991, pages 182, 183, 187 (Ex. 34).

Antti Karttunen and Paul Tek, Table of n, a(n) for n = 0..8191 (First 4096 terms from Karttunen)
J. Lagarias, Products of binomial coefficients and Farey fractions, Lecture in DIMACS Conference on Challenges of Identifying Integer Sequences, October 9, 2014; Part 1, Part 2.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Row sums of triangular table A065040.
Row 1 of array A249421.
Cf. A000295 (a(2^k-2)), A000337 (a(2^k)), A005803 (a(2^k-3)), A036799 (a(2^k+1)), A109363 (a(2^k-4)).
Cf. A000178, A000788, A001142, A002109, A007318, A007814, A174605, A249152, A249150, A249151, A249154, A249343, A249345, A249346, A249347.

a187059 = a007814 . a001142  -- Reinhard Zumkeller, Mar 16 2015

a[n_] := Sum[IntegerExponent[Binomial[n, k], 2], {k, 0, n}]; Array[a, 100, 0]
A187059[n_] := Sum[#*((#+1)*2^k - n - 1) & [Floor[n/2^k]], {k, Floor[Log2[n]]}];
Array[A187059, 100, 0] (* Paolo Xausa, Feb 11 2025 *)
2*Accumulate[#] - Range[Length[#]]*# & [DigitCount[Range[0, 99], 2, 1]] (* Paolo Xausa, Feb 11 2025 *)

a(n)=sum(k=0,n,valuation(binomial(n,k),2))

\\ Much faster version, based on code for A065040 by Charles R Greathouse IV which if reduced even further gives the formula a(n) = 2*A000788(n) - A249154(n):
A065040(m,k) = (hammingweight(k)+hammingweight(m-k)-hammingweight(m));
A187059(n) = sum(k=0, n, A065040(n, k));
for(n=0, 4095, write("b187059.txt", n, " ", A187059(n)));
\\ Antti Karttunen, Oct 25 2014

def A187059(n): return (n+1)*n.bit_count()+sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1)) # Chai Wah Wu, Nov 11 2024

a(2^k-1) = 0 (19th century); a(2^k) = (k-1)*2^k+1 for k >= 1. (Use de Polignac.)
a(n) = Sum_{i=0..n} A065040(n,i) [where the entries of triangular table A065040(m,k) give the exponent of the maximal power of 2 dividing binomial coefficient A007318(m,k)].
a(n) = A007814(A001142(n)). - Jason Kimberley, Nov 02 2011
a(n) = A249152(n) - A174605(n). [Exponent of 2 in the n-th hyperfactorial minus exponent of 2 in the n-th superfactorial. Cf. for example Lagarias & Mehta paper or Peter Luschny's formula for A001142.] - Antti Karttunen, Oct 25 2014
a(n) = 2*A000788(n) - A249154(n). - Antti Karttunen, Nov 02 2014
a(n) = Sum_{i=1..n} (2*i-n-1)*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Jun 02 2022
a(n) = Sum_{k=1..floor(log_2(n))} t*((t+1)*2^k - n - 1), where t = floor(n/(2^k)). - Paolo Xausa, Feb 11 2025, derived from Ridouane Oudra's formula above.

Name clarified by Antti Karttunen, Oct 22 2014

A232599 Alternating sum of cubes, i.e., Sum_{k=0..n} k^p*q^k for p=3, q=-1.

Original entry on oeis.org

0, -1, 7, -20, 44, -81, 135, -208, 304, -425, 575, -756, 972, -1225, 1519, -1856, 2240, -2673, 3159, -3700, 4300, -4961, 5687, -6480, 7344, -8281, 9295, -10388, 11564, -12825, 14175, -15616, 17152, -18785, 20519

Offset: 0

Stanislav Sykora, Nov 26 2013

			a(3) = 0^3 - 1^3 + 2^3 - 3^3 = -20.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A000578 (cubes), A011934 (absolute values), A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

[(1 - (-1)^n*(1 -6*n^2 -4*n^3))/8: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232599:= n-> (1 -(-1)^n*(1 -6*n^2 -4*n^3))/8; seq(A232599(n), n=0..30); # G. C. Greubel, Mar 31 2021

Accumulate[Times@@@Partition[Riffle[Range[0,40]^3,{1,-1},{2,-1,2}],2]] (* Harvey P. Dale, Jul 22 2016 *)

S3M1(n)=((-1)^n*(4*n^3+6*n^2-1)+1)/8;
v = vector(10001);for(k=1,#v,v[k]=S3M1(k-1))

[(1 - (-1)^n*(1 -6*n^2 -4*n^3))/8 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = ((-1)^n*(4*n^3+6*n^2-1) +1)/8.
G.f.: (-x)*(1-4*x+x^2) / ( (1-x)*(1+x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (exp(x) - (1 +10*x -18*x^2 +4*x^3)*exp(-x))/8. - G. C. Greubel, Mar 31 2021
a(n) = - 3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5). - Wesley Ivan Hurt, Mar 31 2021

A089594 Alternating sum of squares to n.

Original entry on oeis.org

-1, 3, -6, 10, -15, 21, -28, 36, -45, 55, -66, 78, -91, 105, -120, 136, -153, 171, -190, 210, -231, 253, -276, 300, -325, 351, -378, 406, -435, 465, -496, 528, -561, 595, -630, 666, -703, 741, -780, 820, -861, 903, -946, 990, -1035, 1081, -1128, 1176, -1225, 1275

Offset: 1

Jon Perry, Dec 30 2003

Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=3, a(n-1)=(-1)^(n-1)*coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 24 2010

Also triangular numbers with alternating signs. - Stanislav Sykora, Nov 26 2013

			a(6) = 1 + 4 - 9 + 16 - 25 + 36 = 3 + 7 + 11 = 21.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), this sequence (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).
Cf. A000217.
Cf. A225144. [Bruno Berselli, Jun 06 2013]

[(-1)^n*n*(n+1)/2: n in [1..50]]; // Vincenzo Librandi, Nov 16 2011

seq(sum(binomial(n,m), m=1..2)-n^2,n=2..51); # Zerinvary Lajos, Jun 19 2008
A089594 := n -> (-1)^n*n*(n+1)/2; # Peter Luschny, Jul 08 2011

nn = Range[50]; Accumulate[(-1)^nn*nn^2] (* Jayanta Basu, Jun 06 2013 *)

for(i=1,50, print1(","sum(j=1,i,(-1)^j*j^2)))

a(n)=(-1)^n*n*(n+1)/2 \\ Charles R Greathouse IV, Jul 08 2011

[(-1)^n*binomial(n+1,2) for n in (1..50)] # G. C. Greubel, Mar 31 2021

From R. J. Mathar, Nov 05 2011: (Start)
a(n) = Sum_{i=1..n} (-1)^i*i^2 = (-1)^n*n*(n+1)/2.
G.f.: -x / (1+x)^3. (End)
a(n) = (-1)^n*det(binomial(i+2,j+1), 1 <= i,j <= n-1). - Mircea Merca, Apr 06 2013
G.f.: -W(0)/(2+2*x), where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) - (k+1)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
E.g.f.: (1/2)*x*(x-2)*exp(-x). - G. C. Greubel, Mar 31 2021
Sum_{n>=1} 1/a(n) = 2 - 4*log(2). - Amiram Eldar, Jan 31 2023

A036800 a(n) = -6 + 2^(n+1)(3 - 2n + n^2).

Original entry on oeis.org

0, 2, 18, 90, 346, 1146, 3450, 9722, 26106, 67578, 169978, 417786, 1007610, 2392058, 5603322, 12976122, 29753338, 67633146, 152567802, 341835770, 761266170, 1686110202, 3716153338, 8153726970, 17817403386, 38788923386

Offset: 0

N. J. A. Sloane

This sequence is a part of a class of sequences of the type: a(n) = sum(i=0,n,(C^i)*(i^k)). This sequence has C=2, k=2. Sequence A036799 has C=2, k=1. Suppose C>=2, k>=1 are integers. What is the general closed form for a(n)? - Ctibor O. Zizka, Feb 07 2008

M. Petkovsek et al., A=B, Peters, 1996, p. 97.
Jolley, Summation of Series, Dover (1961), p. 6.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A232599, A232600, A232601, A232602.

[-6+2^(n+1)*(3-2*n+n^2): n in [0..30]]; // Vincenzo Librandi, Oct 04 2011

A036800:= n-> 2^(n+1)*(3-2*n+n^2) -6; seq(A036800(n), n=0..30); # G. C. Greubel, Mar 31 2021

Table[ -6+2^(n+1)*(3-2*n+n^2),{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2010 *)
LinearRecurrence[{7,-18,20,-8},{0,2,18,90},30] (* Harvey P. Dale, Jun 13 2015 *)

a(n)=2^(n+1)*(3 - 2*n + n^2) - 6 \\ Charles R Greathouse IV, Jun 11 2015

[2^(n+1)*(3-2*n+n^2) -6 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = Sum_{k=0..n} 2^k * k^2. - Benoit Cloitre, Jun 11 2003
From R. J. Mathar, Oct 03 2011: (Start)
G.f.: 2*x*(1+2*x) / ( (1-x)*(1-2*x)^3 ).
a(n) = 2*A036826(n). (End)
a(0)=0, a(1)=2, a(2)=18, a(3)=90, a(n)=7*a(n-1)-18*a(n-2)+ 20*a(n-3)- 8*a(n-4). - Harvey P. Dale, Jun 13 2015
a(n) = Sum_{k=0..n} Sum_{i=0..n} k^2 * C(k,i). - Wesley Ivan Hurt, Sep 21 2017
E.g.f.: 2*(3 -2*x +4*x^2)*exp(2*x) -6*exp(x). - G. C. Greubel, Mar 31 2021

A232600 a(n) = Sum_{k=0..n} k^p*q^k, where p=1, q=-2.

Original entry on oeis.org

0, -2, 6, -18, 46, -114, 270, -626, 1422, -3186, 7054, -15474, 33678, -72818, 156558, -334962, 713614, -1514610, 3203982, -6757490, 14214030, -29826162, 62448526, -130489458, 272163726, -566697074, 1178133390, -2445745266, 5070447502, -10498808946, 21713445774

Offset: 0

Stanislav Sykora, Nov 27 2013

			a(3) = 0^1*2^0 - 1^1*2^1 + 2^1*2^2 - 3^1*2^3 = -18.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-3,0,4).

Cf. A045883, A140960 (absolute values), A059841 (p=0, q=-1), A130472 (p=1 ,q=-1), A089594 (p=2, q=-1), A232599 (p=3, q=-1), A126646 (p=0, q=2), A036799 (p=1, q=2), A036800 (p=q=2), A036827 (p=3, q=2), A077925 (p=0, q=-2), A232601 (p=2, q=-2), A232602 (p=3, q=-2), A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2).

Cf. A045883.

[2*((-2)^n*(3*n+1) -1)/9: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232600:= n-> 2*((-2)^n*(3*n+1) -1)/9; seq(A232600(n), n=0..30); # G. C. Greubel, Mar 31 2021

Table[2((3n+1)(-2)^n -1)/9, {n, 0, 30}] (* Bruno Berselli, Nov 28 2013 *)

PARI
```
a(n)=-((3*n+1)*(-2)^(n+1)+2)/9;
```

[2*((-2)^n*(3*n+1) -1)/9 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = 2*( (3*n+1)*(-2)^n - 1 )/9.
abs(a(n)) = 2*A045883(n) = A140960(n).
From Bruno Berselli, Nov 28 2013: (Start)
G.f.: -2*x / ((1 - x)*(1 + 2*x)^2). [corrected by Georg Fischer, May 11 2019]
a(n) = -3*a(n-1) +4*a(n-3). (End)
From G. C. Greubel, Mar 31 2021: (Start)
E.g.f.: (2/9)*(-exp(x) + (1-6*x)*exp(-2*x)).
a(n) = 2*(-1)^n*A045883(n). (End)

A232601 a(n) = Sum_{k=0..n} k^p*q^k for p = 2 and q = -2.

Original entry on oeis.org

0, -2, 14, -58, 198, -602, 1702, -4570, 11814, -29658, 72742, -175066, 414758, -969690, 2241574, -5131226, 11645990, -26233818, 58700838, -130567130, 288863270, -635980762, 1394062374, -3043511258, 6620165158

Offset: 0

Stanislav Sykora, Nov 27 2013

			a(3) = 0^2*2^0 - 1^2*2^1 + 2^2*2^2 - 3^2*2^3 = -58.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Stanislav Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-5,-6,4,8).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

[2*(1 - (-2)^n*(1-6*n-9*n^2))/27: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232601:= n-> 2*(1 - (-2)^n*(1-6*n-9*n^2))/27; seq(A232601(n), n=0..30); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-5,-6,4,8},{0,-2,14,-58},30] (* Harvey P. Dale, Aug 20 2015 *)

S2M2(n)=((-1)^n*2^(n+1)*(9*n^2+6*n-1)+2)/27;
v = vector(10001); for(k=1, #v, v[k]=S2M2(k-1))

[2*(1 - (-2)^n*(1-6*n-9*n^2))/27 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = 2*((-2)^n * (9*n^2 + 6*n - 1) + 1)/27.
G.f.: 2*x*(-1 + 2*x) / ((1-x)*(1+2*x)^3). - R. J. Mathar, Nov 23 2014
E.g.f.: (2/27)*(exp(x) - (1 +30*x -36*x^2)*exp(-2*x)). - G. C. Greubel, Mar 31 2021
a(n) = - 5*a(n-1) - 6*a(n-2) + 4*a(n-3) + 8*a(n-4). - Wesley Ivan Hurt, Mar 31 2021

A232602 a(n) = Sum_{k=0..n} k^p*q^k, where p=3, q=-2.

Original entry on oeis.org

0, -2, 30, -186, 838, -3162, 10662, -33242, 97830, -275418, 748582, -1977306, 5100582, -12897242, 32060454, -78531546, 189903910, -454052826, 1074770982, -2521320410, 5867287590, -13554437082

Offset: 0

Stanislav Sykora, Nov 27 2013

			a(3) = 0^3*2^0 - 1^3*2^1 + 2^3*2^2 - 3^3*2^3 = -186.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-7,-16,-8,16,16).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

[2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27: n in [0..35]]; // G. C. Greubel, Mar 31 2021

A232602:= n-> 2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27; seq(A232602(n), n=0..35); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-7,-16,-8,16,16}, {0,-2,30,-186,838}, 40] (* G. C. Greubel, Mar 31 2021 *)

a(n)=((-1)^n*2^(n+1)*(27*n^3+27*n^2-9*n-3)+6)/81;

[2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27 for n in (0..35)] # G. C. Greubel, Mar 31 2021

a(n) = 2*(1 - (-2)^n*(1 +3*n -9*n^2 -9*n^3))/27.
G.f.: -2*x*(1-8*x+4*x^2) / ( (1-x)*(1+2*x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (2/27)*(exp(x) - (1 +30*x -144*x^2 +72*x^3)*exp(-2*x)). - G. C. Greubel, Mar 31 2021
a(n) = - 7*a(n-1) - 16*a(n-2) - 8*a(n-3) + 16*a(n-4) + 16*a(n-5). - Wesley Ivan Hurt, Mar 31 2021

cp's OEIS Frontend

A036289 a(n) = n*2^n.

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A000337 a(n) = (n-1)*2^n + 1.

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A130472 A permutation of the integers: a(n) = (-1)^n * floor( (n+1)/2 ).

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A187059 The exponent of highest power of 2 dividing the product of the elements of the n-th row of Pascal's triangle (A001142).

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A232599 Alternating sum of cubes, i.e., Sum_{k=0..n} k^p*q^k for p=3, q=-1.

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A089594 Alternating sum of squares to n.

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A036800 a(n) = -6 + 2^(n+1)(3 - 2n + n^2).