A066185 -id:A066185 - cp's OEIS frontend

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

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

A264034 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A161680(n)) is the number of integer partitions of n with weighted sum k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 3, 2, 1

Offset: 0

Christian Stump, Nov 01 2015

Row sums give A000041.
The weighted sum is given by the sum of the rows where row i is weighted by i.
Note that the first part has weight 0. This statistic (zero-based weighted sum) is ranked by A359677, reverse A359674. Also the number of partitions of n with one-based weighted sum n + k. - Gus Wiseman, Jan 10 2023

			Triangle T(n,k) begins:
  1;
  1;
  1,1;
  1,1,0,1;
  1,1,1,1,0,0,1;
  1,1,1,1,1,0,1,0,0,0,1;
  1,1,1,2,1,0,2,1,0,0,1,0,0,0,0,1;
  1,1,1,2,1,1,2,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1;
  1,1,1,2,2,1,2,2,1,1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0,0,0,0,1;
  ...
The a(15,31) = 5 partitions of 15 with weighted sum 31 are: (6,2,2,1,1,1,1,1), (5,4,1,1,1,1,1,1), (5,2,2,2,2,1,1), (4,3,2,2,2,2), (3,3,3,3,2,1). These are also the partitions of 15 with one-based weighted sum 46. - _Gus Wiseman_, Jan 09 2023

Alois P. Heinz, Rows n = 0..50, flattened
FindStat - Combinatorial Statistic Finder, Weighted size of a partition

Row sums are A000041.
The version for compositions is A053632, ranked by A124757 (reverse A231204).
Row lengths are A152947, or A161680 plus 1.
The one-based version is also A264034, if we use k = n..n(n+1)/2.
The reverse version A358194 counts partitions by sum of partial sums.
A359677 gives zero-based weighted sum of prime indices, reverse A359674.
A359678 counts multisets by zero-based weighted sum.
Cf. A029931, A066185, A124757, A304818, A318283, A337206, A359042.

b:= proc(n, i, w) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, w)+
      `if`(i>n, 0, x^(w*i)*b(n-i, i, w+1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 01 2015

b[n_, i_, w_] := b[n, i, w] = Expand[If[n == 0, 1, If[i < 1, 0, b[n, i - 1, w] + If[i > n, 0, x^(w*i)*b[n - i, i, w + 1]]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Total[Accumulate[Reverse[#]]]==k&]],{n,0,8},{k,n,n*(n+1)/2}] (* Gus Wiseman, Jan 09 2023 *)

From Alois P. Heinz, Jan 20 2023: (Start)
max_{k=0..A161680(n)} T(n,k) = A337206(n).
Sum_{k=0..A161680(n)} k * T(n,k) = A066185(n). (End)

A055580 Björner-Welker sequence: 2^n*(n^2 + n + 2) - 1.

Original entry on oeis.org

1, 7, 31, 111, 351, 1023, 2815, 7423, 18943, 47103, 114687, 274431, 647167, 1507327, 3473407, 7929855, 17956863, 40370175, 90177535, 200278015, 442499071, 973078527, 2130706431, 4647288831, 10099884031, 21877489663

Offset: 0

Wolfdieter Lang, May 26 2000; revised Feb 12 2001

a(n) is the d=1 Betti number of the complement of '3-equal' arrangements in n-dimensional real space, see Björner-Welker reference, Table I, pp. 308-309, column '1' with k=3 and Th. 5.2, pp. 297-298.
Binomial transform of [1/2, 2/3, 3/4, 4/5, ...] = 1/2, 7/6, 31/12, 111/20, 351/30, 1023/42, ..., where 2, 6, 12, 20, ... = A002378 (deleting the zero). - Gary W. Adamson, Apr 28 2005
Number of three-dimensional block structures associated with n joint systems in the construction of stable underground structures. - Richard M. Green, Jul 26 2011
Number of monotone mappings from the chain with three points to the complete binary tree of height n (n+1 levels). For example, the seven monotone mappings from the chain with three points (denoted 1,2,3, in order) to the complete binary tree with two levels (with a the root of the tree, and b, c the atoms) are: f(1)=f(2)=f(3)=a; f(1)=f(2)=a, f(3)=b; f(1)=f(2)=a, f(3)=c; f(1)=a, f(2)=f(3)=b; f(1)=a, f(2)=f(3)=c; f(1)=f(2)=f(3)=b; f(1)=f(2)=f(3)=c. - Pietro Codara, Mar 26 2015

H. Barcelo and S. Smith, The discrete fundamental group of the order complex of B_n, Abstract 1020-05-141, 1020th Meeting Amer. Math. Soc., Cincinatti, Ohio, Oct 21-22, 2006.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Henry Adams, Samir Shukla, and Anurag Singh, Čech complexes of hypercube graphs, arXiv:2212.05871 [math.CO], 2022.
H. Barcelo and R. Laubenbacher, Perspectives on A-homotopy theory and its applications, Discr. Math., 298 (2005), 39-61.
H. Barcelo and S. Smith, The discrete fundamental group of the order complex of B_n, arXiv:0711.0915 [math.CO], 2007.
A. Björner and V. Welker, The homology of "k-equal" manifolds and related partition lattices, Adv. Math., 110 (1995), 277-313.
Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
G.G. Kocharyan and A.M. Kulyukin, Construction of a three-dimensional block structure on the basis of jointed rock parameters estimating the stability of underground workings, Soil Mech. Found. Eng., 31 (1994), 62-66.
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 (7,-18,20,-8).

Fourth column of triangle A055252.

Cf. A055252, A055249, A045618, A000337, A001788, A066185.

[2^n*(n^2+n+2)-1: n in [0..35]]; // Vincenzo Librandi, Jul 28 2011

Table[ n*(n+1)*2^(n-2), {n, 0, 26}] // Accumulate // Rest (* Jean-François Alcover, Jul 09 2013, after Paul Barry *)
LinearRecurrence[{7,-18,20,-8},{1,7,31,111},30] (* Harvey P. Dale, Nov 27 2014 *)

a(n)=(n^2+n+2)<Charles R Greathouse IV, Jul 28 2011

a(n) = A055252(n+3, 3).
a(n) = Sum_{j=0..n-1} a(j) + A045618(n), n >= 1.
G.f.: 1/((1-2*x)^3*(1-x)).
Partial sums of A001788 (without leading zero). - Paul Barry, Jun 26 2003
a(n) = A001788(n) - A000337(n). - Jon Perry, Dec 12 2003
a(n) = A119258(n+4,n). - Reinhard Zumkeller, May 11 2006
E.g.f.: 2*(1 + 2*x + 2*x^2)*exp(2*x) - exp(x). - G. C. Greubel, Oct 28 2016
a(n) = Sum_{k=0..n+1} Sum_{i=0..n+1} i^2 * C(k,i). - Wesley Ivan Hurt, Sep 21 2017

Edited (for consistency with change of offset) by M. F. Hasler, Nov 03 2012

cp's OEIS Frontend

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

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A264034 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A161680(n)) is the number of integer partitions of n with weighted sum k.

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A055580 Björner-Welker sequence: 2^n*(n^2 + n + 2) - 1.

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A066184 Sum of the first moments of all partitions of n with weight starting at 1.

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