A011377 -id:A011377 - cp's OEIS frontend

A013609 Triangle of coefficients in expansion of (1+2*x)^n.

1, 1, 2, 1, 4, 4, 1, 6, 12, 8, 1, 8, 24, 32, 16, 1, 10, 40, 80, 80, 32, 1, 12, 60, 160, 240, 192, 64, 1, 14, 84, 280, 560, 672, 448, 128, 1, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 1, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512, 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024

Offset: 0

N. J. A. Sloane

T(n,k) is the number of lattice paths from (0,0) to (n,k) with steps (1,0) and two kinds of steps (1,1). The number of paths with steps (1,0) and s kinds of steps (1,1) corresponds to the expansion of (1+s*x)^n. - Joerg Arndt, Jul 01 2011
Also sum of rows in A046816. - Lior Manor, Apr 24 2004
Also square array of unsigned coefficients of Chebyshev polynomials of second kind. - Philippe Deléham, Aug 12 2005
The rows give the number of k-simplices in the n-cube. For example, 1, 6, 12, 8 shows that the 3-cube has 1 volume, 6 faces, 12 edges and 8 vertices. - Joshua Zucker, Jun 05 2006
Triangle whose (i, j)-th entry is binomial(i, j)*2^j.
With offset [1,1] the triangle with doubled numbers, 2*a(n,m), enumerates sequences of length m with nonzero integer entries n_i satisfying sum(|n_i|) <= n. Example n=4, m=2: [1,3], [3,1], [2,2] each in 2^2=4 signed versions: 2*a(4,2) = 2*6 = 12. The Sum over m (row sums of 2*a(n,m)) gives 2*3^(n-1), n >= 1. See the W. Lang comment and a K. A. Meissner reference under A024023. - Wolfdieter Lang, Jan 21 2008
n-th row of the triangle = leftmost column of nonzero terms of X^n, where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (2,2,2,...) in the subdiagonal. - Gary W. Adamson, Jul 19 2008
Numerators of a matrix square-root of Pascal's triangle A007318, where the denominators for the n-th row are set to 2^n. - Gerald McGarvey, Aug 20 2009
From Johannes W. Meijer, Sep 22 2010: (Start)
The triangle sums (see A180662 for their definitions) link the Pell-Jacobsthal triangle, whose mirror image is A038207, with twenty-four different sequences; see the crossrefs.
This triangle may very well be called the Pell-Jacobsthal triangle in view of the fact that A000129 (Kn21) are the Pell numbers and A001045 (Kn11) the Jacobsthal numbers.
(End)
T(n,k) equals the number of n-length words on {0,1,2} having n-k zeros. - Milan Janjic, Jul 24 2015
T(n-1,k-1) is the number of 2-compositions of n with zeros having k positive parts; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
T(n,k) is the number of chains 0=x_0Geoffrey Critzer, Oct 01 2022
Excluding the initial 1, T(n,k) is the number of k-faces of a regular n-cross polytope. See A038207 for n-cube and A135278 for n-simplex. - Mohammed Yaseen, Jan 14 2023

			Triangle begins:
  1;
  1,  2;
  1,  4,   4;
  1,  6,  12,    8;
  1,  8,  24,   32,   16;
  1, 10,  40,   80,   80,    32;
  1, 12,  60,  160,  240,   192,    64;
  1, 14,  84,  280,  560,   672,   448,    128;
  1, 16, 112,  448, 1120,  1792,  1792,   1024,    256;
  1, 18, 144,  672, 2016,  4032,  5376,   4608,   2304,    512;
  1, 20, 180,  960, 3360,  8064, 13440,  15360,  11520,   5120,  1024;
  1, 22, 220, 1320, 5280, 14784, 29568,  42240,  42240,  28160, 11264,  2048;
  1, 24, 264, 1760, 7920, 25344, 59136, 101376, 126720, 112640, 67584, 24576, 4096;
From _Peter Bala_, Apr 20 2012: (Start)
The triangle can be written as the matrix product A038207*(signed version of A013609).
  |.1................||.1..................|
  |.2...1............||-1...2..............|
  |.4...4...1........||.1..-4...4..........|
  |.8..12...6...1....||-1...6...-12...8....|
  |16..32..24...8...1||.1..-8....24.-32..16|
  |..................||....................|
(End)

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
G. Hotz, Zur Reduktion von Schaltkreispolynomen im Hinblick auf eine Verwendung in Rechenautomaten, El. Datenverarbeitung, Folge 5 (1960), pp. 21-27.

T. D. Noe, Rows n=0..50 of triangle, flattened
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
H. J. Brothers, Pascal's Prism: Supplementary Material, PDF version.
John Cartan, Cartan's triangle shows the relationship to the n-cube.
R. Ehrenborg and M. Readdy, Characterization of the factorial functions of Eulerian binomial and Sheffer posets
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients of k rooks on the 2xn board, all heights 2.
W. G. Harter, Representations of multidimensional symmetries in networks, J. Math. Phys., 15 (1974), 2016-2021.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
D. A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 666 (2017), 21-35.

Cf. A007318, A013610, etc.
Appears in A167580 and A167591. - Johannes W. Meijer, Nov 23 2009
From Johannes W. Meijer, Sep 22 2010: (Start)
Triangle sums (see the comments): A000244 (Row1); A000012 (Row2); A001045 (Kn11); A026644 (Kn12); 4*A011377 (Kn13); A000129 (Kn21); A094706 (Kn22); A099625 (Kn23); A001653 (Kn3); A007583 (Kn4); A046717 (Fi1); A007051 (Fi2); A077949 (Ca1); A008998 (Ca2); A180675 (Ca3); A092467 (Ca4); A052942 (Gi1); A008999 (Gi2); A180676 (Gi3); A180677 (Gi4); A140413 (Ze1); A180678 (Ze2); A097117 (Ze3); A055588 (Ze4).
(End)
Cf. A024023, A038207, A046816, A105728, A115068, A135278, A171977, A180662.
T(2n,n) gives A059304.

a013609 n = a013609_list !! n
a013609_list = concat $ iterate ([1,2] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

a013609 n k = a013609_tabl !! n !! k
a013609_row n = a013609_tabl !! n
a013609_tabl = iterate (\row -> zipWith (+) ([0] ++ row) $
                                zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Jul 22 2013, Feb 27 2013

[2^k*Binomial(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 17 2021

bin2:=proc(n,k) option remember; if k<0 or k>n then 0 elif k=0 then 1 else 2*bin2(n-1,k-1)+bin2(n-1,k); fi; end; # N. J. A. Sloane, Jun 01 2009

Flatten[Table[CoefficientList[(1 + 2*x)^n, x], {n, 0, 10}]][[1 ;; 59]] (* Jean-François Alcover, May 17 2011 *)
BinomialROW[n_, k_, t_] := Sum[Binomial[n, k]*Binomial[k, j]*(-1)^(k - j)*t^j, {j, 0, k}]; Column[Table[BinomialROW[n, k, 3], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jan 28 2019 *)

a(n,k):=coeff(expand((1+2*x)^n),x^k);
create_list(a(n,k),n,0,6,k,0,n); /* Emanuele Munarini, Nov 21 2012 */

/* same as in A092566 but use */
steps=[[1,0], [1,1], [1,1]]; /* note double [1,1] */
/* Joerg Arndt, Jul 01 2011 */

flatten([[2^k*binomial(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Sep 17 2021

G.f.: 1 / (1 - x*(1+2*y)).
T(n,k) = 2^k*binomial(n,k).
T(n,k) = 2*T(n-1,k-1) + T(n-1,k). - Jon Perry, Nov 22 2005
Row sums are 3^n = A000244(n). - Joerg Arndt, Jul 01 2011
T(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i). - Mircea Merca, Apr 28 2012
E.g.f.: exp(2*y*x + x). - Geoffrey Critzer, Nov 12 2012
Riordan array (x/(1 - x), 2*x/(1 - x)). Exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(1 + 6*x + 12*x^2/2! + 8*x^3/3!) = 1 + 8*x + 40*x^2/2! + 160*x^3/3! + 560*x^4/4! + .... The same property holds more generally for Riordan arrays of the form (f(x), 2*x/(1 - x)). - Peter Bala, Dec 21 2014
T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(n,k) * binomial(k,j) * 3^j. - Kolosov Petro, Jan 28 2019
T(n,k) = 2*(n+1-k)*T(n,k-1)/k, T(n,0) = 1. - Alexander R. Povolotsky, Oct 08 2023
For n >= 1, GCD(T(n,1), ..., T(n,n)) = GCD(T(n,1),T(n,n)) = GCD(2*n,2^n) = A171977(n). - Pontus von Brömssen, Nov 01 2024

A178420 Partial sums of floor(2^n/3).

Original entry on oeis.org

0, 1, 3, 8, 18, 39, 81, 166, 336, 677, 1359, 2724, 5454, 10915, 21837, 43682, 87372, 174753, 349515, 699040, 1398090, 2796191, 5592393, 11184798, 22369608, 44739229, 89478471, 178956956, 357913926, 715827867, 1431655749, 2863311514

Offset: 1

Mircea Merca, Dec 21 2010

Essentially the same as A011377: 0 followed by the terms of A011377. - Joerg Arndt, Apr 22 2016

Partial sums of A000975(n-1).

			a(5) = 0 + 1 + 2 + 5 + 10 = 18.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Column k=2 of A368296.

Cf. A000975, A178455.

[Floor((4*2^n-3*n-4)/6): n in [1..30]]; // Vincenzo Librandi, Jun 23 2011

Maple
```
seq(round((4*2^n-3*n-4)/6),n=1..50)
```

f[n_] := Floor[(4 2^n - 3 n - 4)/6]; f[Range[60]] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2011 *)
CoefficientList[Series[x / ((1 + x) (1 - 2 x) (1 - x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
LinearRecurrence[{3,-1,-3,2},{0,1,3,8},40] (* or *) Accumulate[ Table[ Floor[ 2^n/3],{n,40}]] (* Harvey P. Dale, Dec 24 2015 *)

a(n)=(4<Charles R Greathouse IV, Jul 31 2013

a(n) = A011377(n-1) for n >= 1. - Joerg Arndt, Apr 22 2016
a(n) = round((8*2^n - 6*n - 9)/12).
a(n) = floor((4*2^n - 3*n - 4)/6).
a(n) = ceiling((4*2^n - 3*n - 5)/6).
a(n) = round((4*2^n - 3*n - 4)/6).
a(n) = a(n-2) + 2^(n-1) - 1, n > 2.
From Bruno Berselli, Jan 15 2011: (Start)
a(n) = (8*2^n - 6*n - 9 + (-1)^n)/12.
G.f.: x^2/((1+x)*(1-2*x)*(1-x)^2). (End)
G.f.: Q(0)/(3*(1-x)^2), where Q(k) = 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
a(n) = 2*a(n-1) + floor(n/2) for n > 1. - Bruno Berselli, Apr 30 2014
a(n) = floor(2^(n+1)/3) - floor((n+1)/2). - Seiichi Manyama, Dec 22 2023

A166753 Partial sums of A166752.

Original entry on oeis.org

1, 2, 5, 6, 17, 18, 61, 62, 233, 234, 917, 918, 3649, 3650, 14573, 14574, 58265, 58266, 233029, 233030, 932081, 932082, 3728285, 3728286, 14913097, 14913098, 59652341, 59652342, 238609313, 238609314, 954437197, 954437198, 3817748729, 3817748730

Offset: 0

Paul Barry, Oct 21 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-4,4).

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x-2*x^2-4*x^3)/((1-x)*(1-5*x^2+4*x^4)) )); // G. C. Greubel, Jun 06 2019

LinearRecurrence[{1,5,-5,-4,4}, {1,2,5,6,17}, 40] (* G. C. Greubel, May 24 2016 *)
Accumulate[LinearRecurrence[{0,5,0,-4},{1,1,3,1},40]] (* Harvey P. Dale, Aug 12 2024 *)

my(x='x+O('x^40)); Vec((1+x-2*x^2-4*x^3)/((1-x)*(1-5*x^2+4*x^4))) \\ G. C. Greubel, Sep 30 2017

((1+x-2*x^2-4*x^3)/((1-x)*(1-5*x^2+4*x^4))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 06 2019

G.f.: (1+x-2*x^2-4*x^3)/((1-x)*(1-5*x^2+4*x^4)).
a(n) = a(n+1) + 5*a(n+2) - 5*a(n-3) - 4*a(n-4) + 4*a(n-5).
a(n) = (4/3)*A061547(n+1) - (1/3)*A166754(n).
a(n) = (4/3)*A061547(n+1) - (1/3)*A000975(n) + (4/3)*A011377(n-2).

A091597 Triangle read by rows: T(n,0) = A001045(n+1), T(n,n)=1, T(n,m) = T(n-1,m-1) + T(n-1,m).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 5, 5, 3, 1, 11, 10, 8, 4, 1, 21, 21, 18, 12, 5, 1, 43, 42, 39, 30, 17, 6, 1, 85, 85, 81, 69, 47, 23, 7, 1, 171, 170, 166, 150, 116, 70, 30, 8, 1, 341, 341, 336, 316, 266, 186, 100, 38, 9, 1, 683, 682, 677, 652, 582, 452, 286, 138, 47, 10, 1

Offset: 0

Paul Barry, Jan 23 2004

A Jacobsthal-Pascal triangle.

Equals triangle M * Pascal's triangle, M = an infinite lower triangular Toeplitz matrix with A078008: [1, 0, 2, 2, 6, 10, 22, 42, ...] in every column. - Gary W. Adamson, May 25 2009

			Triangle begins as:
    1;
    1,   1;
    3,   2,   1;
    5,   5,   3,   1;
   11,  10,   8,   4,   1;
   21,  21,  18,  12,   5,   1;
   43,  42,  39,  30,  17,   6,   1;
   85,  85,  81,  69,  47,  23,   7,  1;
  171, 170, 166, 150, 116,  70,  30,  8, 1;
  341, 341, 336, 316, 266, 186, 100, 38, 9, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Columns include A001045, A000975, A011377.
Row sums are A059570.
Cf. A078008. - Gary W. Adamson, May 25 2009

Flat(List([0..12], n->List([0..n], k-> Sum([0..n], j-> 2^j*Binomial(n-j, k+j)) ))); # G. C. Greubel, Jun 04 2019

[[(&+[2^j*Binomial(n-j, k+j): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 04 2019

A091597 := proc(n,k)
    if k = 0 then
        A001045(n+1) ;
    elif k = n then
        1 ;
    elif k <0 or k > n then
        0 ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Oct 05 2012

Table[Sum[Binomial[n-j, k+j]*2^j, {j,0,n}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 04 2019 *)

{T(n,k) = sum(j=0, n, 2^j*binomial(n-j, k+j))}; \\ G. C. Greubel, Jun 04 2019

[[sum(2^j*binomial(n-j, k+j) for j in (0..n)) for k in (0..n)] for n in [0..12]] # G. C. Greubel, Jun 04 2019

Number triangle: T(n, k) = Sum_{j=0..n} binomial(n-j, k+j)2^j.
Riordan array: (1/(1-x-2*x^2), x/(1-x)).
k-th column has g.f. (1/(1-x-2*x^2))*(x/(1-x))^k.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - 2*T(n-3,k) - 2*T(n-3,k-1), T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=3, T(2,1)=2, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 11 2014

A130127 Triangle defined by A000012 * A130125, read by rows.

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 2, 4, 4, 8, 3, 4, 8, 8, 16, 3, 6, 8, 16, 16, 32, 4, 6, 12, 16, 32, 32, 64, 4, 8, 12, 24, 32, 64, 64, 128, 5, 8, 16, 24, 48, 64, 128, 128, 256, 5, 10, 16, 32, 48, 96, 128, 256, 256, 512, 6, 10, 20, 32, 64, 96, 192, 256, 512, 512, 1024, 6, 12, 20, 40, 64, 128, 192, 384, 512, 1024, 1024, 2048

Offset: 1

Gary W. Adamson, May 11 2007

Row sums = A011377: (1, 3, 8, 18, 39, ...). A130126 = A130125 * A000012.

			First few rows of the triangle:
  1;
  1, 2;
  2, 2,  4;
  2, 4,  4,  8;
  3, 4,  8,  8, 16;
  3, 6,  8, 16, 16, 32;
  4, 6, 12, 16, 32, 32, 64;
  ...

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A000012, A130125, A130126, A011377.

[[2^(k-1)*Floor((n-k+2)/2): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 06 2019

Table[2^(k-1)*Floor[(n-k+2)/2], {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, Jun 06 2019 *)

{T(n,k) = 2^(k-1)*floor((n-k+2)/2)}; \\ G. C. Greubel, Jun 06 2019

[[2^(k-1)*floor((n-k+2)/2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 06 2019

T(n,k) = 2^(k-1) * floor((n-k+2)/2). - G. C. Greubel, Jun 06 2019

More terms added by G. C. Greubel, Jun 06 2019

A166754 a(n) = 4A061547(n+1) - 3A166753(n).

Original entry on oeis.org

1, 2, 9, 22, 53, 114, 241, 494, 1005, 2026, 4073, 8166, 16357, 32738, 65505, 131038, 262109, 524250, 1048537, 2097110, 4194261, 8388562, 16777169, 33554382, 67108813, 134217674, 268435401, 536870854, 1073741765, 2147483586

Offset: 0

Paul Barry, Oct 21 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A061547, A166753.

Cf. A000975, A011377.

List([0..40], n-> (2^(n+3) + (-1)^n - (4*n+7))/2) # G. C. Greubel, Jun 04 2019

[(2^(n+3) +(-1)^n -(4*n+7))/2: n in [0..40]]; // G. C. Greubel, Oct 10 2017

LinearRecurrence[{3,-1,-3,2}, {1,2,9,22}, 40] (* G. C. Greubel, May 24 2016 *)

my(x='x+O('x^40)); Vec((1-x+4*x^2)/((1+x)*(1-x)^2*(1-2*x))) \\ G. C. Greubel, Oct 10 2017

[(2^(n+3) + (-1)^n - (4*n+7))/2 for n in (0..40)] # G. C. Greubel, Jun 04 2019

G.f.: (1-x+4*x^2)/((1+x)*(1-x)^2*(1-2*x)).
a(n) = (2^(n+3) + (-1)^n - (4*n+7))/2.
a(n) = A000975(n) - 4*A011377(n-2).
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).
E.g.f.: (8*exp(2*x) + exp(-x) - (4*x+7)*exp(x))/2. - G. C. Greubel, Jun 04 2019

A272144 Convolution of A000217 and A001045.

Original entry on oeis.org

0, 0, 1, 4, 12, 30, 69, 150, 316, 652, 1329, 2688, 5412, 10866, 21781, 43618, 87300, 174672, 349425, 698940, 1397980, 2796070, 5592261, 11184654, 22369452, 44739060, 89478289, 178956760, 357913716, 715827642, 1431655509, 2863311258, 5726622772

Offset: 0

Patrick Okolo Edeogu, Apr 21 2016

			a(4) = 12 = 0*10+1*6+1*3+3*1+5*0 from A000217: 0,1,3,6,10,... and A001045: 0,1,1,3,5,11,...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,5,-2).

Partial Sums of A011377(n-2)=A178420(n-1).

m:=40; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^2/((1-x)^3*(1+x)*(1-2*x)))); // G. C. Greubel, Oct 26 2018

seq(coeff(series(x^2/((1-x)^3*(1+x)*(1-2*x)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 26 2018

CoefficientList[Series[x^2/((1 - x)^3 (1 + x) (1 - 2 x)), {x, 0, 30}], x] (* Michael De Vlieger, Apr 21 2016 *)

concat([0, 0], Vec(x^2/((1-x)^3*(1+x)*(1-2*x)) + O(x^40))) \\ Altug Alkan, Apr 21 2016

a(n) = Sum{k=0..n} A000217(k) * A001045(n-k). - Joerg Arndt, May 17 2016
a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 5*a(n-4) - 2*a(n-5).
G.f.: x^2/((1-x)^3*(1+x)*(1-2*x)).
a(n+2) = (-105+(-1)^n+2^(7+n)-48*n-6*n^2)/24. - Colin Barker, Apr 21 2016
E.g.f.: (exp(-x) + 32*exp(2*x) - 3*(11 + 10*x + 2*x^2)*exp(x))/24. - Ilya Gutkovskiy, Apr 21 2016

A320933 a(n) = 2^n - floor((n+3)/2).

Original entry on oeis.org

0, 0, 2, 5, 13, 28, 60, 123, 251, 506, 1018, 2041, 4089, 8184, 16376, 32759, 65527, 131062, 262134, 524277, 1048565, 2097140, 4194292, 8388595, 16777203, 33554418, 67108850, 134217713, 268435441, 536870896

Offset: 0

Paul Curtz, Oct 28 2018

The sequence 0, 0, a(n) is an autosequence of the second kind. The difference table is:
   0,   0,   0,   0,   2,   5,  13, ...
   0,   0,   0,   2,   3,   8,  15, ...
   0,   0,   2,   1,   5,   7,  17, ...
   0,   2,  -1,   4,   2,  10,  14, ...
   2,  -3,   5,  -2,   8,   4,  20, ...
  -5,   8,  -7,  10,  -4,  16,   8, ...
  13, -15,  17, -14,  20,  -8,  32, ...
etc.

Colin Barker, Table of n, a(n) for n = 0..1000
OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A000079, A004526, A011377, A014551, A060182, A166920.

List([0..40],n->2^n-Int((n+3)/2)); # Muniru A Asiru, Oct 28 2018

[((-1)^n+2^(n+2)-2*n-5)/4: n in [0..40]]; // G. C. Greubel, Jun 04 2019

seq(2^n-floor((n+3)/2),n=0..40); # Muniru A Asiru, Oct 28 2018

a[n_]:=2^n - Floor[(n+3)/2]; Array[a, 40, 0] (* or *) CoefficientList[ Series[x^2*(2-x)/((1-x)^2*(1-x-2*x^2)), {x, 0, 40}], x] (* Stefano Spezia, Oct 28 2018 *)

concat([0,0], Vec(x^2*(2-x)/((1-x)^2*(1+x)*(1-2*x)) + O(x^40))) \\ Colin Barker, Oct 28 2018

[((-1)^n+2^(n+2)-2*n-5)/4 for n in (0..40)] # G. C. Greubel, Jun 04 2019

a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4).
a(n+1) = a(n) + A166920(n).
a(n+4) - a(n) = 13, 28, 58, 118, ... = 15*2^n - 2 = A060182(n+2).
With b(n) = 0, 0, 0, A011377(n) = 0, 0, 0, 1, 3, 8, 18, ..., then a(n) = 2*b(n+1) - b(n).
a(n+2) - 2*a(n+1) + a(n) = A014551(n).
G.f.: x^2*(2 - x)/((1-x)^2*(1 - x - 2*x^2)). - Stefano Spezia, Oct 28 2018
a(n) = ((-1)^n + 2^(n+2) - 2*n - 5) / 4. - Colin Barker, Oct 28 2018

Three terms corrected by Colin Barker, Oct 28 2018

A091598 Triangle read by rows: T(n,0) = A078008(n), T(n,m) = T(n-1,m-1) + T(n-1,m).

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 2, 3, 2, 1, 6, 5, 5, 3, 1, 10, 11, 10, 8, 4, 1, 22, 21, 21, 18, 12, 5, 1, 42, 43, 42, 39, 30, 17, 6, 1, 86, 85, 85, 81, 69, 47, 23, 7, 1, 170, 171, 170, 166, 150, 116, 70, 30, 8, 1, 342, 341, 341, 336, 316, 266, 186, 100, 38, 9, 1, 682, 683, 682, 677, 652, 582, 452, 286, 138, 47, 10, 1

Offset: 0

Paul Barry, Jan 23 2004

A Jacobsthal-Pascal triangle.

			Triangle starts as:
   1;
   0,  1;
   2,  1,  1;
   2,  3,  2,  1;
   6,  5,  5,  3,  1;
  10, 11, 10,  8,  4,  1;
  22, 21, 21, 18, 12,  5, 1;
  42, 43, 42, 39, 30, 17, 6, 1; ...

G. C. Greubel, Rows n = 0..20 of triangle, flattened

Columns include A078008, A001045, A000975, A011377. Row sums give A084219.

Cf. A091597.

T[n_, k_]:= If[k==0, (2^n + 2*(-1)^n)/3, If[k<0 || k>n, 0, T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 04 2019 *)

{T(n,k) = if(k==0, (2^n + 2*(-1)^n)/3, if(k<0 || k>n, 0, T(n-1,k-1) + T(n-1,k)))}; \\ G. C. Greubel, Jun 04 2019

def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return (2^n + 2*(-1)^n)/3
    else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jun 04 2019

k-th column has e.g.f. ((1-x)/(1-x-x^2))*(x/(1-x))^k.

cp's OEIS Frontend

A013609 Triangle of coefficients in expansion of (1+2*x)^n.

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A178420 Partial sums of floor(2^n/3).

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A166753 Partial sums of A166752.

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A091597 Triangle read by rows: T(n,0) = A001045(n+1), T(n,n)=1, T(n,m) = T(n-1,m-1) + T(n-1,m).

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A130127 Triangle defined by A000012 * A130125, read by rows.

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A166754 a(n) = 4*A061547(n+1) - 3*A166753(n).

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A166754 a(n) = 4A061547(n+1) - 3A166753(n).