A000295 -id:A000295 - cp's OEIS frontend

A014741 Numbers k such that k divides 2^(k+1) - 2.

1, 2, 6, 18, 42, 54, 126, 162, 294, 342, 378, 486, 882, 1026, 1134, 1314, 1458, 1806, 2058, 2394, 2646, 3078, 3402, 3942, 4374, 5334, 5418, 6174, 6498, 7182, 7938, 9198, 9234, 10206, 11826, 12642, 13122, 14154, 14406, 16002, 16254

Offset: 1

N. J. A. Sloane, Olivier Gérard

nonn

Also, numbers k such that k divides Eulerian number A000295(k+1) = 2^(k+1) - k - 2.
Also, numbers k such that k divides A086787(k) = Sum_{i=1..k} Sum_{j=1..k} i^j.
All terms greater than 1 are even; for a proof, see comment in A036236. - Max Alekseyev, Feb 03 2012
If k>1 is a term, then 3*k is also a term. - Alexander Adamchuk, Nov 03 2006
Prime numbers of the form a(m)+1 are given by A069051. - Max Alekseyev, Nov 14 2012
The number 2^m - 2 is a term of this sequence if and only if m - 1 is a term. - Thomas Ordowski, Jul 01 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000295, A086787, A015919, A006517, A330382.

Join[{1,2},Select[Range[17000],PowerMod[2,#+1,#]==2&]] (* Harvey P. Dale, Feb 11 2015 *)

is(n)=Mod(2,n)^(n+1)==2 \\ Charles R Greathouse IV, Nov 03 2016

For n > 1, a(n) = 2*A014945(n-1). - Max Alekseyev, Nov 14 2012

A035042 a(n) = 2^n - C(n,0)- ... - C(n,9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 79, 378, 1471, 4944, 14893, 41226, 106762, 262144, 616666, 1401292, 3096514, 6690448, 14198086, 29703676, 61450327, 126025204, 256737233, 520381366, 1050777737, 2115862624, 4251885323, 8531819446

Offset: 0

N. J. A. Sloane

J. Eckhoff, Der Satz von Radon in konvexen Productstrukturen II, Monat. f. Math., 73 (1969), 7-30.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

a(n)= A055248(n, 10). Partial sums of A035041.
Cf. A000079, A000225, A000295, A002663, A002664, A035038-A035041.
Cf. A007318.

a035042 n = a035042_list !! n
a035042_list = map (sum . drop 10) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015

a:=n->sum(binomial(n,j),j=10..n): seq(a(n), n=0..33); # Zerinvary Lajos, Jan 04 2007

Table[2^n-Sum[Binomial[n,i],{i,0,9}],{n,0,40}] (* Harvey P. Dale, Jan 05 2013 *)

G.f.: x^10/((1-2*x)*(1-x)^10).

A006231 a(n) = Sum_{k=2..n} n(n-1)...(n-k+1)/k.

Original entry on oeis.org

0, 1, 5, 20, 84, 409, 2365, 16064, 125664, 1112073, 10976173, 119481284, 1421542628, 18348340113, 255323504917, 3809950976992, 60683990530208, 1027542662934897, 18430998766219317, 349096664728623316, 6962409983976703316, 145841989688186383337

Offset: 1

R. K. Guy

a(n) is also the number of permutations in the symmetric group S_n that are pure cycles, see example. - Avi Peretz (njk(AT)netvision.net.il), Mar 24 2001
Also the number of elementary circuits in a complete directed graph with n nodes [D. B. Johnson, 1975]. - N. J. A. Sloane, Mar 24 2014
If one takes 1,2,3,4, ..., n and starts creating parenthetic products of k-tuples and adding, one gets a(n+1). For 1,2,3,4 one gets (1)+(2)+(3)+(4) = 10; (1*2)+(2*3)+(3*4) = 20; (1*2*3)+(2*3*4) = 30; (1*2*3*4) = 24; and 10+20+30+24 = 84 = a(5). - J. M. Bergot, Apr 24 2014
Let P_n be the set of probability distributions over orderings of n objects that can be obtained by drawing n real numbers from independent probability distributions and sorting. Then a(n) is conjectured to be the dimension of P_n, as a semi-algebraic subset of R^(n!). - Jamie Tucker-Foltz, Jul 29 2024

			a(3) = 5 because the cycles in S_3 are (12), (13), (23), (123), (132).
a(4) = 20 because there are 24 permutations of {1,2,3,4} but we don't count (12)(34), (13)(24), (14)(23) or the identity permutation. - _Geoffrey Critzer_, Nov 03 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 100 terms from T. D. Noe)
Eric Babson, Moon Duchin, Annina Iseli, Pietro Poggi-Corradini, Dylan Thurston, Jamie Tucker-Foltz, Models of Random Spanning Trees, arXiv:2407.20226 [math.CO], 2023.
R. K. Guy, Letter to N. J. A. Sloane, 1977
Donald B. Johnson, Finding all the elementary circuits of a directed graph, SIAM J. Comput. 4 (1975), 77-84. MR0398155 (53 #2010).
Jamie Tucker-Foltz, Code to compute dimension of P_n on GitHub.

Cf. A059760, A000295.

Column k=1 of A136394.

a006231 n = numerator $
   sum $ tail $ zipWith (%) (scanl1 (*) [n,(n-1)..1]) [1..n]
-- Reinhard Zumkeller, Dec 27 2011

A006231 := proc(n)
    n*( hypergeom([1,1,1-n],[2],-1)-1) ;
    simplify(%) ;
end proc: # R. J. Mathar, Aug 06 2013

a[n_] = n*(HypergeometricPFQ[{1,1,1-n}, {2}, -1] - 1); Table[a[n], {n, 1, 20}] (* Jean-François Alcover,  Mar 29 2011 *)
Table[Sum[Times@@Range[n-k+1,n]/k,{k,2,n}],{n,20}] (* Harvey P. Dale, Sep 23 2011 *)

a(n) = n--; sum(ip=1, n, sum(j=1, n-ip+1, prod(k=j, j+ip-1, k))); \\ Michel Marcus, May 07 2014 after comment by J. M. Bergot

a(n+1) - a(n) = A000522(n) - 1.
a(n) = n*( 3F1(1,1,1-n; 2;-1) -1). - Jean-François Alcover,  Mar 29 2011
E.g.f.: exp(x)*(log(1/(1-x))-x). - Geoffrey Critzer, Sep 12 2012
G.f.: (Q(0) - 1)/(1-x)^2, where Q(k)= 1 + (2*k + 1)*x/( 1 - x - 2*x*(1-x)*(k+1)/(2*x*(k+1) + (1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 09 2013
Conjecture: a(n) + (-n-2)*a(n-1) + (3*n-2)*a(n-2) + 3*(-n+2)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Aug 06 2013

More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001

A055356 Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 2, 0, 1, 11, 18, 6, 0, 1, 26, 98, 96, 24, 0, 1, 57, 424, 874, 600, 120, 0, 1, 120, 1614, 6040, 8244, 4320, 720, 0, 1, 247, 5682, 35458, 83500, 83628, 35280, 5040, 0, 1, 502, 19022, 187288, 701164, 1169768, 915984, 322560, 40320, 0, 1

Offset: 1

Christian G. Bower, May 15 2000

In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.

Also related to the solution of the equation df/dt=f e^f (see the Maple code). - F. Chapoton, Jul 16 2004

			Triangle begins
  1;
  1,  0;
  1,  1,   0;
  1,  4,   2,   0;
  1, 11,  18,   6,   0;
  1, 26,  98,  96,  24,   0;
  1, 57, 424, 874, 600, 120, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - _N. J. A. Sloane_, Aug 21 2012
Index entries for sequences related to mobiles

Row sums give A029768 (p(n,1)).
Alternating row sums give A089963 (p(n+1,-1)).
Columns 2..8 are A000295, A055357, A055358, A055359, A055360, A055361, A055362.
Cf. A055340, A055349, A055363.

P[1]:=1;for n from 1 to 8 do P[n+1]:=simplify((1+n*x)*P[n]+x*diff(P[n],x)) end; # F. Chapoton, Jul 16 2004

P[1][_] = 1;
P[n_][x_] := P[n][x] = (1 + (n-1) x) P[n-1][x] + x P[n-1]'[x] // Expand;
row[1] = {1};
row[n_] := Append[CoefficientList[P[n-1][x], x], 0];
Array[row, 10] // Flatten (* Jean-François Alcover, Nov 17 2018, after F. Chapoton *)

A(n)={my(v=vector(n)); v[1]=y; for(n=2, #v, v[n]=v[n-1] + sum(k=1, n-2, binomial(n-2, k)*v[k]*v[n-k])); vector(#v, i, Vecrev(v[i]/y, i))}
{ my(T=A(10)); for(i=1, #T, print(T[i])) } \\ Andrew Howroyd, Sep 23 2018

Let p(n,x) be the polynomial with coefficients equal to the n-th row of the triangle in ascending powers of x, e.g., p(4,x) = 1+4*x+2*x^2; then p(n+1,x) = (1+(n-1)*x)*p(n,x) + x*p'(n,x). - Ben Whitmore, May 12 2021
Recurrence: T(n,k) = (n-2) * T(n-1,k-1) + k * T(n-1,k) for n >= 1, 1 <= k <= n with T(1,1) = 1 and T(n,k) = 0 for n < 1, k < 1 or k > n. - Georg Fischer, Oct 27 2021
Conjecture: row polynomials are R(n-2,0) for n > 1 where R(n,k) = R(n-1,k+1) + x*Sum_{i=0..n-1} Sum_{j=0..k} binomial(n-1, i)*R(n-i-1,j)*R(i,k-j) for n > 0, k >= 0 with R(0,k) = 1 for k >= 0. - Mikhail Kurkov, Apr 11 2025

A087903 Triangle read by rows of the numbers T(n,k) (n > 1, 0 < k < n) of set partitions of n of length k which do not have a proper subset of parts with a union equal to a subset {1,2,...,j} with j < n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 11, 9, 1, 1, 26, 48, 16, 1, 1, 57, 202, 140, 25, 1, 1, 120, 747, 916, 325, 36, 1, 1, 247, 2559, 5071, 3045, 651, 49, 1, 1, 502, 8362, 25300, 23480, 8260, 1176, 64, 1, 1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1, 1, 2036, 82509, 525608, 998830, 749154, 253764, 40944, 3105, 100, 1

Offset: 2

Mike Zabrocki, Oct 14 2003

Another version of the triangle T(n,k), 0 <= k <= n, given by [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938; see also A086329 for a triangle transposed. - Philippe Deléham, Jun 13 2004

			T(2,1)=1 for {12};
T(3,1)=1, T(3,2) = 1 for {123}; {13|2};
T(4,1)=1, T(4,2)=4, T(4,3)=1 for {1234}; {14|23}, {13|24}, {124|3}, {134|2}; {14|2|3}.
From _Philippe Deléham_, Jul 16 2007: (Start)
Triangle begins:
  1;
  1,    1;
  1,    4,     1;
  1,   11,     9,      1;
  1,   26,    48,     16,      1;
  1,   57,   202,    140,     25,     1;
  1,  120,   747,    916,    325,    36,     1;
  1,  247,  2559,   5071,   3045,   651,    49,    1;
  1,  502,  8362,  25300,  23480,  8260,  1176,   64,  1;
  1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1;
  ...
Triangle T(n,k), 0 <= k <= n, given by [1,0,2,0,3,0,4,0,...] DELTA [0,1,0,1,0,1,0,...] begins:
  1;
  1,    0;
  1,    1,     0;
  1,    4,     1,      0;
  1,   11,     9,      1,      0;
  1,   26,    48,     16,      1,     0;
  1,   57,   202,    140,     25,     1,     0;
  1,  120,   747,    916,    325,    36,     1,    0;
  1,  247,  2559,   5071,   3045,   651,    49,    1,  0;
  1,  502,  8362,  25300,  23480,  8260,  1176,   64,  1, 0;
  1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1, 0;
  ...
(End)

G. C. Greubel, Rows n = 2..52 of the triangle, flattened
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
Mercedes H. Rosas and Bruce E. Sagan, Symmetric functions in noncommuting variables, arXiv:math/0208168 [math.CO], 2002, 2004.
Yves Le Jan, Loop clusters on complete graphs, arXiv:2504.16976 [math.PR], 2025. See p. 6.
Margarete C. Wolf, Symmetric Functions of Non-commutative Elements, Duke Math. J., 2 (1936), 626-637.

Cf. A000110, A000295, A008277, A055106, A074664.

Cf. A055105.

A := proc(n,k) option remember; local j,ell; if n<=0 or k>=n then 0; elif k=1 or k=n-1 then 1; else S2(n-1,k)+add(add((k-ell-1)*A(n-j-1,k-ell)*S2(j,ell),ell=0..k-1),j=0..n-2); fi; end: S2 := (n,k)->if n<0 or k>n then 0; elif k=n or k=1 then 1 else k*S2(n-1,k)+S2(n-1,k-1); fi:

nmax = 12; t[n_, k_] := t[n, k] = StirlingS2[n-1, k] + Sum[ (k-d-1)*t[n-j-1, k-d]*StirlingS2[j, d], {d, 0, k-1}, {j, 0, n-2}]; Flatten[ Table[ t[n, k], {n, 2, nmax}, {k, 1, n-1}]] (* Jean-François Alcover, Oct 04 2011, after given formula *)

@CachedFunction # T = A087903
def T(n,k): return stirling_number2(n-1, k) + sum( sum( (k-m-1)*T(n-j-1, k-m)*stirling_number2(j, m) for m in (0..k-1) ) for j in (0..n-2) )
flatten([[T(n, k) for k in (1..n-1)] for n in (2..14)]) # G. C. Greubel, Jun 21 2022

T(n, n-1) = T(n,1) = 1.
T(n, n-2) = (n-2)^2.
T(n, 2) = A000295(n).
T(n, k) = S2(n-1, k) + Sum_{j=0..n-2} Sum_{d=0..k-1} (k-d-1)*T(n-j-1, k-d)*S2(j, d), where S2(n, k) is the Stirling number of the second kind.
Sum_{k = 1..n-1} T(n, k) = A074664(n). - Philippe Deléham, Jun 13 2004
G.f.: 1-1/(1+add(add(q^n t^k S2(n, k), k=1..n), n >= 1)) where S2(n, k) are the Stirling numbers of the 2nd kind A008277. - Mike Zabrocki, Sep 03 2005

A242351 Number T(n,k) of isoscent sequences of length n with exactly k ascents; triangle T(n,k), n>=0, 0<=k<=n+3-ceiling(2*sqrt(n+2)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 11, 3, 1, 26, 25, 1, 57, 128, 17, 1, 120, 525, 229, 2, 1, 247, 1901, 1819, 172, 1, 502, 6371, 11172, 3048, 53, 1, 1013, 20291, 58847, 33065, 2751, 7, 1, 2036, 62407, 280158, 275641, 56905, 1422, 1, 4083, 187272, 1242859, 1945529, 771451, 61966, 436

Offset: 0

Joerg Arndt and Alois P. Heinz, May 11 2014

An isoscent sequence of length n is an integer sequence [s(1),...,s(n)] with s(1) = 0 and 0 <= s(i) <= 1 plus the number of level steps in [s(1),...,s(i)].
Columns k=0-10 give: A000012, A000295, A243228, A243229, A243230, A243231, A243232, A243233, A243234, A243235, A243236.
Row sums give A000110.
Last elements of rows give A243237.

			T(4,0) = 1: [0,0,0,0].
T(4,1) = 11: [0,0,0,1], [0,0,0,2], [0,0,0,3], [0,0,1,0], [0,0,1,1], [0,0,2,0], [0,0,2,1], [0,0,2,2], [0,1,0,0], [0,1,1,0], [0,1,1,1].
T(4,2) = 3: [0,0,1,2], [0,1,0,1], [0,1,1,2].
Triangle T(n,k) begins:
  1;
  1;
  1,    1;
  1,    4;
  1,   11,     3;
  1,   26,    25;
  1,   57,   128,    17;
  1,  120,   525,   229,     2;
  1,  247,  1901,  1819,   172;
  1,  502,  6371, 11172,  3048,   53;
  1, 1013, 20291, 58847, 33065, 2751, 7;
  ...

Joerg Arndt and Alois P. Heinz, Rows n = 0..100, flattened

Cf. A048993 (for counting level steps), A242352 (for counting descents), A137251 (ascent sequences counting ascents), A238858 (ascent sequences counting descents), A242153 (ascent sequences counting level steps), A083479.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 0$2)):
seq(T(n), n=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n<1, 1, Expand[Sum[If[j>i, x, 1]*b[n-1, j, t + If[j == i, 1, 0]], {j, 0, t+1}]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n-1, 0, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 09 2015, after Maple *)

A232603 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=2, q=-1/2.

Original entry on oeis.org

0, -1, 2, -5, 6, -13, 10, -29, 6, -69, -38, -197, -250, -669, -1142, -2509, -4762, -9813, -19302, -38965, -77530, -155501, -310518, -621565, -1242554, -2485733, -4970790, -9942309, -19883834, -39768509, -79536118

Offset: 0

Stanislav Sykora, Nov 27 2013

The factor 2^n (i.e., |1/q|^n) is present to keep the values integer.

See also A232600 and references therein for integer values of q.

			a(3) = 2^3 * [0^2/2^0 - 1^2/2^1 + 2^2/2^2 - 3^2/2^3] = -5.

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 (-1,3,5,2).

Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232604 (p=3,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).

[((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232603:= n-> ((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27; seq(A232603(n), n=0..35); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-1,3,5,2}, {0,-1,2,-5}, 35] (* G. C. Greubel, Mar 31 2021 *)

a(n)=((-1)^n*(9*n^2+12*n+2)-2^(n+1))/27;

[((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = ((-1)^n*(9*n^2+12*n+2) - 2^(n+1))/27.
G.f.: x*(-1+x)/( (1-2*x)*(1+x)^3 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (1/27)*(-2*exp(2*x) + (2 -21*x +9*x^2)*exp(-x)). - G. C. Greubel, Mar 31 2021
a(n) = - a(n-1) + 3*a(n-2) + 5*a(n-3) + 2*a(n-4). - Wesley Ivan Hurt, Mar 31 2021

A232604 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=3, q=-1/2.

Original entry on oeis.org

0, -1, 6, -15, 34, -57, 102, -139, 234, -261, 478, -375, 978, -241, 2262, 1149, 6394, 7875, 21582, 36305, 80610, 151959, 314566, 616965, 1247754, 2479883, 4977342, 9935001, 19891954, 39759519, 79546038, 159062285

Offset: 0

Stanislav Sykora, Nov 27 2013

The factor 2^n (i.e., |1/q|^n) is present to make the values integers.
See also A232600 and references therein for integer values of q.
The same values with different signs are produced by a(n) = n^3 - 2*a(n). The signs are all positive until n = 15, with negative signs on values for all subsequent odd indices. - Richard R. Forberg, Feb 17 2014.

			a(3) = 2^3 * (0^3/2^0 - 1^3/2^1 + 2^3/2^2 - 3^3/2^3) = 0-4+16-27 = -15.

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 (-2,2,8,7,2).

Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232603 (p=2,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).

[(2^(n+1) + (-1)^n*(9*n^3 +18*n^2 +6*n -2))/27: n in [0..35]]; // G. C. Greubel, Mar 31 2021

A232604:= n-> (2^(n+1) +(-1)^n*(9*n^3 +18*n^2 +6*n -2))/27; seq(A232604(n), n=0..30); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-2,2,8,7,2}, {0,-1,6,-15,34}, 35] (* G. C. Greubel, Mar 31 2021 *)

a(n)=(2^(n+1)+(-1)^n*(9*n^3+18*n^2+6*n-2))/27;

[(2^(n+1) + (-1)^n*(9*n^3 +18*n^2 +6*n -2))/27 for n in (0..35)] # G. C. Greubel, Mar 31 2021

a(n) = (2^(n+1) + (-1)^n*(9*n^3+18*n^2+6*n-2))/27.
G.f.: x*(1-4*x+x^2) / ( (2*x-1)*(1+x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (1/27)*(2*exp(2*x) - (2 +33*x -45*x^2 +9*x^3)*exp(-x)). - G. C. Greubel, Mar 31 2021
a(n) = - 2*a(n-1) + 2*a(n-2) + 8*a(n-3) + 7*a(n-4) + 2*a(n-5). - Wesley Ivan Hurt, Mar 31 2021

A016208 Expansion of 1/((1-x)(1-3x)(1-4x)).

Original entry on oeis.org

1, 8, 45, 220, 1001, 4368, 18565, 77540, 320001, 1309528, 5326685, 21572460, 87087001, 350739488, 1410132405, 5662052980, 22712782001, 91044838248, 364760483725, 1460785327100, 5848371485001, 23409176469808, 93683777468645, 374876324642820, 1499928942876001

Offset: 0

N. J. A. Sloane

Binomial transform of A085277. - Paul Barry, Jun 25 2003

Number of walks of length 2n+5 between two nodes at distance 5 in the cycle graph C_12. - Herbert Kociemba, Jul 05 2004

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Natalia Agudelo Muñetón, Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, and Isaías David Marín Gaviria, Brauer Configuration Algebras and Their Applications in Graph Energy Theory, Mathematics (2021) Vol. 9, 3042.
Index entries for linear recurrences with constant coefficients, signature (8,-19,12)

Cf. A000225, A000295, A000392, A002275, A003462, A003463, A003464, A023000, A023001, A002452, A016123, A016125, A016256.

a:=[1,8,45];; for n in [4..30] do a[n]:=8*a[n-1]-19*a[n-2]+12*a[n-3]; od; Print(a); # Muniru A Asiru, Apr 19 2019

Table[(2^(2*n + 3) - 3^(n + 2) + 1)/6, {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[1/((1-x)(1-3x)(1-4x)),{x,0,30}],x] (* or *) LinearRecurrence[ {8,-19,12},{1,8,45},30] (* Harvey P. Dale, Apr 09 2012 *)

Vec(1/((1-x)*(1-3*x)*(1-4*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = 16*4^n/3 + 1/6 - 9*3^n/2. - Paul Barry, Jun 25 2003
a(0) = 0, a(1) = 8, a(n) = 7*a(n-1) - 12*a(n-2) + 1. - Vincenzo Librandi, Feb 10 2011
a(0) = 1, a(1) = 8, a(2) = 45, a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3). - Harvey P. Dale, Apr 09 2012

A035039 a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,6).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 9, 46, 176, 562, 1586, 4096, 9908, 22819, 50643, 109294, 230964, 480492, 988116, 2014992, 4084248, 8243109, 16587165, 33308926, 66794952, 133820134, 267936278, 536249296, 1072973612, 2146540999

Offset: 0

N. J. A. Sloane

Partial sums of A035038.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
J. Eckhoff, Der Satz von Radon in konvexen Produktstrukturen II, Monat. f. Math., 73 (1969), 7-30.
Ângela Mestre, José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (9,-35,77,-105,91,-49,15,-2).

Cf. A000079, A000225, A000295, A002663, A002664, A035038-A035042.

Cf. A007318.

a035039 n = a035039_list !! n
a035039_list = map (sum . drop 7) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015

a:=n->sum(binomial(n,j),j=7..n): seq(a(n), n=0..31); # Zerinvary Lajos, Feb 12 2007

a=1;lst={};s1=s2=s3=s4=s5=s6=s7=0;Do[s1+=a;s2+=s1;s3+=s2;s4+=s3;s5+=s4;s6+=s5;s7+=s6;AppendTo[lst,s7];a=a*2,{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)
Table[2^n-Total[Binomial[n,Range[0,6]]],{n,40}] (* or *) LinearRecurrence[ {9,-35,77,-105,91,-49,15,-2},{0,0,0,0,0,0,0,1},40] (* Harvey P. Dale, Apr 22 2016 *)

a(n) = A055248(n,7).
G.f.: x^7/((1-2*x)*(1-x)^7).
a(n) = Sum_{k=0..n}, C(n, k+7) = Sum_{k=7..n} C(n, k); a(n) = 2a(n-1) + C(n-1, 6). - Paul Barry, Aug 23 2004

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A014741 Numbers k such that k divides 2^(k+1) - 2.

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A035042 a(n) = 2^n - C(n,0)- ... - C(n,9).

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A006231 a(n) = Sum_{k=2..n} n(n-1)...(n-k+1)/k.

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A055356 Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.

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A087903 Triangle read by rows of the numbers T(n,k) (n > 1, 0 < k < n) of set partitions of n of length k which do not have a proper subset of parts with a union equal to a subset {1,2,...,j} with j < n.

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A242351 Number T(n,k) of isoscent sequences of length n with exactly k ascents; triangle T(n,k), n>=0, 0<=k<=n+3-ceiling(2*sqrt(n+2)), read by rows.

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A232603 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=2, q=-1/2.

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A016208 Expansion of 1/((1-x)(1-3x)(1-4x)).