A045883 -id:A045883 - cp's OEIS frontend

A094705 Convolution of Jacobsthal(n) and 3^n.

0, 1, 4, 15, 50, 161, 504, 1555, 4750, 14421, 43604, 131495, 395850, 1190281, 3576304, 10739835, 32241350, 96767741, 290390604, 871346575, 2614389250, 7843866801, 23532998504, 70601791715, 211810967550, 635444087461, 1906354632004, 5719108635255, 17157415384250

Offset: 0

Paul Barry, May 21 2004

For k>2, a(n,k)=k^(n+1)/((k-2)(k+1))-2^(n+1)/(3k-6)-(-1)^n/(3k+3) gives the convolution of Jacobsthal(n) and k^n.

In general x/((1-ax)(1-ax-bx^2)) expands to Sum_{k=0..floor(n/2)} C(n-k,k+1)a^(n-k-1)*(b/a)^k. - Paul Barry, Oct 25 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-6).

Cf. A001045 (Jacobsthal), A000244(3^n), A045883.

[(3^(n+2) -2^(n+3) -(-1)^n)/12: n in [0..50]]; // G. C. Greubel, Jul 21 2022

LinearRecurrence[{4,-1,-6},{0,1,4},30] (* Harvey P. Dale, Apr 02 2017 *)
Jacob0[n_] := (2^n - (-1)^n)/3; a[n_] := First@ListConvolve[Table[Jacob0[i], {i, 0, n}], 3^Range[0, n]]; Table[a[x], {x, 0, 10}] (* Robert P. P. McKone, Nov 28 2020 *)

concat(0, Vec(x/((1+x)*(1-2*x)*(1-3*x)) + O(x^50))) \\ Michel Marcus, Sep 13 2014

[(3^(n+1) - lucas_number1(n+3, 1, -2))/4 for n in (0..50)] # G. C. Greubel, Jul 21 2022

G.f.: x/((1+x)*(1-2*x)*(1-3*x)).
a(n) = (3^(n+2) - 2^(n+3) - (-1)^n)/12.
a(n) = 4*a(n-1) -a(n-2) -6*a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k+1)*2^(n-k-1)*(3/2)^k. - Paul Barry, Oct 25 2004
a(n) = (3^(n+1) - A001045(n+3))/4. - G. C. Greubel, Jul 21 2022

A124763 Number of non-rises (levels or falls) for compositions in standard order.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 2, 0, 1, 1, 3, 0, 1, 1, 2, 0, 2, 1, 3, 0, 1, 1, 2, 1, 2, 2, 4, 0, 1, 1, 2, 1, 2, 1, 3, 0, 1, 2, 3, 1, 2, 2, 4, 0, 1, 1, 2, 0, 2, 1, 3, 1, 2, 2, 3, 2, 3, 3, 5, 0, 1, 1, 2, 1, 2, 1, 3, 0, 2, 2, 3, 1, 2, 2, 4, 0, 1, 1, 2, 1, 3, 2, 4, 1, 2, 2, 3, 2, 3, 3, 5, 0, 1, 1, 2, 1, 2, 1, 3, 0

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. a(n) is one fewer than the number of maximal strictly increasing runs in this composition. Alternatively, a(n) is the number of weak descents in the same composition. For example, the strictly increasing runs of the 1234567th composition are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)), so a(1234567) = 8 - 1 = 7. The 7 weak descents together with the strict ascents are: 3 >= 2 >= 1 < 2 >= 2 >= 1 < 2 < 5 >= 1 >= 1 >= 1. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; 2>=1>=1, so a(11) = 2.
The table starts:
  0
  0
  0 1
  0 1 0 2
  0 1 1 2 0 1 1 3
  0 1 1 2 0 2 1 3 0 1 1 2 1 2 2 4
  0 1 1 2 1 2 1 3 0 1 2 3 1 2 2 4 0 1 1 2 0 2 1 3 1 2 2 3 2 3 3 5

Cf. A029931, A066099, A124760, A124761, A124764, A011782 (row lengths), A045883 (row sums), A238343, A333220.
Compositions of n with k weak descents are A333213.
Positions of zeros are A333255.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Partial sums from the right are A048793.
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Select[Partition[stc[n],2,1],GreaterEqual@@#&]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1=b(i+1)} 1.
a(n) = A124761(n) + A124762(n).
For n > 0, a(n) = A124768(n) - 1. - Gus Wiseman, Apr 08 2020

A094953 Triangle T(n,m) read by rows: number of rises (drops) in the compositions of n with m parts, m>=2.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 2, 8, 9, 4, 3, 12, 21, 16, 5, 3, 18, 39, 44, 25, 6, 4, 24, 66, 96, 80, 36, 7, 4, 32, 102, 184, 200, 132, 49, 8, 5, 40, 150, 320, 430, 372, 203, 64, 9, 5, 50, 210, 520, 830, 888, 637, 296, 81, 10, 6, 60, 285, 800, 1480, 1884, 1673, 1024, 414, 100, 11, 6

Offset: 2

Ralf Stephan, May 26 2004

			1
1 2
2 4 3
2 8 9 4
3 12 21 16 5
3 18 39 44 25 6
4 24 66 96 80 36 7

S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.

Columns 2-4 (+-offset) are A004526, A007590, A007518.
Row sums are A045883, diagonals include n, n^2, (n-1)(n^2-n+2)/2, (n-1)^2(n^+n+6), etc.
Cf. A045927.

T[n_, m_] := SeriesCoefficient[(m-1)x^(m+1)/(1+x)/(1-x)^m, {x, 0, n+1}];
Table[T[n, m], {n, 2, 13}, {m, 2, n}] // Flatten (* Jean-François Alcover, Dec 03 2018 *)

T(n,m)=polcoeff((m-1)*x^(m+1)/(1+x)/(1-x)^m,n)

G.f. of m-th column: [(m-1)x^(m+1)]/[(1+x)(1-x)^m].

A102301 a(n) = ((3n + 1)2^(n+3) + 9 + (-1)^n)/18.

Original entry on oeis.org

1, 4, 13, 36, 93, 228, 541, 1252, 2845, 6372, 14109, 30948, 67357, 145636, 313117, 669924, 1427229, 3029220, 6407965, 13514980, 28428061, 59652324, 124897053, 260978916, 544327453, 1133394148, 2356266781, 4891490532, 10140895005, 20997617892, 43426891549

Offset: 0

Creighton Dement, Feb 20 2005

A floretion-generated sequence resulting from particular transform of A000975.

Floretion Algebra Multiplication Program, FAMP Code: 2jesforseq[ + .5'i + 'kk' + .5'jk' ], 1vesforseq(n) = A000975(n+2)*(-1)^(n+1), ForType: 1A, LoopType: tes (2nd iteration)

G. C. Greubel, Table of n, a(n) for n = 0..1000
T. Etzion, On the stopping redundancy of Reed-Muller codes, IEEE Trans. Information Theory 52 (11) (2006) 4867-4879, also, arXiv:cs/0511056 [cs.IT], 2005.
Toufik Mansour and Armend Sh. Shabani, Bargraphs in bargraphs, Turkish Journal of Mathematics (2018) Vol. 42, Issue 5, 2763-2773.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,4).

Cf. A000975, A000337, A045883, A001787, A059570, A137266, A008574, A059260.

[((3*n+1)*2^(n+3)+9+(-1)^n)/18: n in [0..40]]; // Vincenzo Librandi, Nov 21 2018

Table[((3n+1)*2^(n+3) + 9 + (-1)^n)/18, {n,0,50}] (* G. C. Greubel, Sep 27 2017 *)
LinearRecurrence[{4, -3, -4, 4}, {1, 4, 13, 36}, 50] (* Vincenzo Librandi, Nov 21 2018 *)

a(n)=((3*n+1)*2^(n+3)+9+(-1)^n)/18 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: 1/((1-x^2)*(1-2*x)^2).
a(n+1) - 2*a(n) = A000975(n+2) (n-th number without consecutive equal binary digits)
a(n) + a(n+1) = A000337(n+2);
a(n+1) - a(n) = A045883(n+2);
a(n+2) - a(n) = A001787(n+3) ( Number of edges in n-dimensional hypercube );
a(n+2) - 2*a(n+1) + a(n) = A059570(n+3);
Convolution of "Number of fixed points in all 231-avoiding involutions in S_n" (A059570) with the natural numbers (A000027), treating the result as if offset=0. - Graeme McRae, Jul 12 2006
Equals triangle A059260 * A008574 as a vector, where A008574 = [1, 4, 8, 12, 16, 20, ...]. - Gary W. Adamson, Mar 06 2012

A113861 a(n) = (1/9)((6n - 7)*2^(n-1) - (-1)^n).

Original entry on oeis.org

0, 1, 5, 15, 41, 103, 249, 583, 1337, 3015, 6713, 14791, 32313, 70087, 151097, 324039, 691769, 1470919, 3116601, 6582727, 13864505, 29127111, 61050425, 127693255, 266571321, 555512263, 1155763769, 2401006023, 4980969017, 10319851975, 21355531833, 44142719431

Offset: 1

N. J. A. Sloane, Jan 25 2006

This sequence is connected with the Collatz problem (see the sequences A045883 and A001045). - Michel Lagneau, Jan 13 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
T. Etzion, On the stopping redundancy of Reed-Muller codes, IEEE Trans. Information Theory 52 (11) (2006) 4867-4879; arXiv:cs.IT/0511056.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A102301.

Cf. A059260, A016813.

Join[{0},Numerator[CoefficientList[Series[(x+1)/((x-1)*(x^2+x-2)),{x,0,40}],x]]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)
LinearRecurrence[{3,0,-4},{0,1,5},40] (* Harvey P. Dale, Jun 16 2022 *)

a(n)=((6*n-7)<<(n-1)-(-1)^n)/9 \\ Charles R Greathouse IV, Jan 13 2012

a(n+1) - 2*a(n) = A001045(n+2), Jacobsthal numbers. - Paul Curtz, Jul 05 2008
3*a(n) - a(n+1) = -1, -2, 4*a(n). - Paul Curtz, Jul 05 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: x^2*(1+2*x)/((1+x)*(1-2*x)^2).
a(n) + a(n+1) = A014480(n-1). (End)
a(n) = 4*a(n-1) - 4*a(n-2) + (-1)^(n+1), n>2. - Gary Detlefs, Dec 19 2010
a(n) = 3*a(n-1) - 4*a(n-3), n>3. - Gary Detlefs, Dec 19 2010
a(n) = n*2^n - A045883(n). - Michel Lagneau, Jan 13 2012
Starting with "1" = triangle A059260 * A016813 as a vector, where A016813 = (4n + 1): [ 1, 5, 9, 13, ...]. - Gary W. Adamson, Mar 06 2012

A137241 Number triples (k,3-k,2-2k), concatenated for k=0, 1, 2, 3,...

Original entry on oeis.org

0, 3, 2, 1, 2, 0, 2, 1, -2, 3, 0, -4, 4, -1, -6, 5, -2, -8, 6, -3, -10, 7, -4, -12, 8, -5, -14, 9, -6, -16, 10, -7, -18, 11, -8, -20, 12, -9, -22, 13, -10, -24, 14, -11, -26, 15, -12, -28, 16, -13, -30, 17, -14, -32, 18, -15, -34, 19, -16, -36, 20, -17, -38, 21, -18, -40

Offset: 0

Paul Curtz, Mar 09 2008

The entries are the coefficients in a family of Jacobsthal recurrences: a(n)=k*a(n-1)+(3-k)*a(n-2)+(2-2k)*a(n-3).
Examples for k=0 are in A001045 and A113954. Examples for k=1 are A001045, A078008.
Examples for k=2 are A000975, A087288, A084639, A000012 and A001045.
Examples for k=3 are A045883, A059570. Examples for k=4 are A094705 and A015518.

			The triples (k,3-k,2-2k) are (0,3,2), (1,2,0), (2,1,-2), (3,0,-4),...

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

CoefficientList[Series[x*(3 + 2*x + x^2 - 4*x^3 - 4*x^4)/((x - 1)^2*(1 + x + x^2)^2), {x, 0, 50}], x] (* G. C. Greubel, Sep 28 2017 *)
Table[{n,3-n,2-2n},{n,0,30}]//Flatten (* or *) LinearRecurrence[ {0,0,2,0,0,-1},{0,3,2,1,2,0},100] (* Harvey P. Dale, Jun 23 2019 *)

x='x+O('x^50); Vec(x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x +x^2 )^2)) \\ G. C. Greubel, Sep 28 2017

From R. J. Mathar, Feb 25 2009: (Start)
a(n) = 2*a(n-3) - a(n-6).
G.f.: x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x+x^2)^2). (End)

Edited by R. J. Mathar, Jun 28 2008

A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

Original entry on oeis.org

0, 1, 0, -1, 2, 0, 3, -2, 4, 0, -5, 6, -4, 8, 0, 11, -10, 12, -8, 16, 0, -21, 22, -20, 24, -16, 32, 0, 43, -42, 44, -40, 48, -32, 64, 0, -85, 86, -84, 88, -80, 96, -64, 128, 0, 171, -170, 172, -168, 176, -160, 192, -128, 256, 0, -341, 342, -340, 344, -336, 352, -320, 384, -256, 512, 0

Offset: 0

Paul Curtz, Jul 24 2008

A variant of the triangle A140503, now including the diagonal.

Since the diagonal contains zeros, rows sums are those of A140503.

			Triangle begins as:
    0;
    1,   0;
   -1,   2,   0;
    3,  -2,   4,  0;
   -5,   6,  -4,  8,   0;
   11, -10,  12, -8,  16,  0;
  -21,  22, -20, 24, -16, 32,  0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A001045, A016131, A045883, A052992, A083086, A084175.

Cf. A091056, A097038, A109765, A115489, A140503, A192382, A246036.

[2^k*(1-(-2)^(n-k))/3: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 18 2023

A001045:= n -> (2^n-(-1)^n)/3;
A140944:= proc(n,k) if n = 0 then A001045(k); else procname(n-1,k+1)-procname(n-1,k) ; fi; end:
seq(seq(A140944(n,k),k=0..n),n=0..10); # R. J. Mathar, Sep 07 2009

T[0, 0]=0; T[1, 0]= T[0, 1]= 1; T[0, k_]:= T[0, k]= T[0, k-1] + 2*T[0, k-2]; T[n_, n_]=0; T[n_, k_]:= T[n, k] = T[n-1, k+1] - T[n-1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Dec 17 2014 *)
Table[2^k*(1-(-2)^(n-k))/3, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2023 *)

T(n, k) = (2^k - 2^n*(-1)^(n+k))/3 \\ Jianing Song, Aug 11 2022

def A140944(n,k): return 2^k*(1 - (-2)^(n-k))/3
flatten([[A140944(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Feb 18 2023

T(n, k) = T(n-1, k+1) - T(n-1, k). T(0, k) = A001045(k).
T(n, k) = (2^k - 2^n*(-1)^(n+k))/3, for n >= k >= 0. - Jianing Song, Aug 11 2022
From G. C. Greubel, Feb 18 2023: (Start)
T(n, n-1) = A000079(n).
T(2*n, n) = (-1)^(n+1)*A192382(n+1).
T(2*n, n-1) = (-1)^n*A246036(n-1).
T(2*n, n+1) = A083086(n).
T(3*n, n) = -A115489(n).
Sum_{k=0..n} T(n, k) = A052992(n)*[n>0] + 0*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A045883(n).
Sum_{k=0..n} 2^k*T(n, k) = A084175(n).
Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^(n+1)*A109765(n).
Sum_{k=0..n} 3^k*T(n, k) = A091056(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*A097038(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^(n+1)*A138495(n). (End)

Edited and extended by R. J. Mathar, Sep 07 2009

A140960 a(n) = (2(-1)^n - 2^(n+1) + 3n*2^n)/9.

Original entry on oeis.org

0, 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, 44858547314

Offset: 0

Paul Curtz, Jul 26 2008

Specify that a triangle has T(n,0) = T(n,n) = A001045(n), and T(r,c) = T(r-1,c-1) + T(r-1,c). The sum of the terms in the first n rows is a(n+1). - J. M. Bergot, May 21 2013

a(n) is the difference between the total number of runs of equal parts in the compositions of n+1, and the compositions of n+1. - Gregory L. Simay, May 04 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-4)

[( 2*(-1)^n-2^(n+1)+3*n*2^n)/9: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011

LinearRecurrence[{3,0,-4},{0,0,2},40] (* Harvey P. Dale, Apr 14 2015 *)

a(n)=(2*(-1)^n-2^(n+1)+3*n*2^n)/9 \\ Charles R Greathouse IV, Oct 16 2015

a(n+1) - 2*a(n) = A078008(n+1) = 2*A001045(n).
G.f.: 2*x^2/((1+x)*(1-2*x)^2).
a(n) = 2*A045883(n-1).
a(n) = 3*a(n-1) - 4*a(n-3), n > 2.
a(n) = A059570(n+1) - A011782(n+1). - Gregory L. Simay, May 04 2017

Definition replaced with Lava's closed form of August 2008 by R. J. Mathar, Feb 11 2010

A140503 Triangle T(d,n) read by rows, the n-th term of the d-th differences of the Jacobsthal sequence A001045.

Original entry on oeis.org

1, -1, 2, 3, -2, 4, -5, 6, -4, 8, 11, -10, 12, -8, 16, -21, 22, -20, 24, -16, 32, 43, -42, 44, -40, 48, -32, 64, -85, 86, -84, 88, -80, 96, -64, 128, 171, -170, 172, -168, 176, -160, 192, -128, 256, -341, 342, -340, 344, -336, 352, -320, 384, -256, 512, 683, -682, 684, -680

Offset: 1

Paul Curtz, Jun 30 2008

If interpreted as a flat sequence a(j), we obtain a(j+1)-2a(j)= -3, 4, -1, -8, 8, -13, 16, -16, 16, -5, -32, 32, -32, 32, -53, 64, ... which is essentially the negative values of A096773 padded by groups of one, then two, then three etc. signed elements of A098354.

			A001045 and its d times iterated differences are
.0,.1,.1,.3,.5,11,21,43,...
.1,.0,.2,.2,.6,10,22,... < d=1
-1,.2,.0,.4,.4,12,... < d=2
.3,-2,.4,.0,.8,.. < d=3
-5,.6,-4,.8,.0,...
The sequence contains the first d elements of the d-th row, those up to the diagonal (which contains zeros).

Cf. A001045, A140944 (with an extra diagonal of 0's).

T(d,n) = (2^n - 2^d*(-1)^(d+n))/3 \\ Jianing Song, Aug 11 2022

T(d,n)=T(d-1,n+1)-T(d-1,n). T(0,n)=A001045(n).
Row sums: sum_{n=0..d-1} T(d,n) = A002450([(d+1)/2]).
Row sums of absolute values: sum_{n=0..d-1} |T(d,n)| = A045883(d).
T(d,n) = (2^n - 2^d*(-1)^(d+n))/3, for d > n >= 0. - Jianing Song, Aug 11 2022

Edited by R. J. Mathar, Jul 14 2008

A246788 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x+2)^k.

Original entry on oeis.org

1, -3, 2, 9, -10, 3, -23, 38, -21, 4, 57, -122, 99, -36, 5, -135, 358, -381, 204, -55, 6, 313, -986, 1299, -916, 365, -78, 7, -711, 2598, -4077, 3564, -1875, 594, -105, 8, 1593, -6618, 12051, -12564, 8205, -3438, 903, -136, 9, -3527, 16422, -34029, 41196, -32115, 16722, -5817, 1304, -171, 10

Offset: 0

Derek Orr, Nov 15 2014

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x+2)^0 + A_1*(x+2)^1 + A_2*(x+2)^2 + ... + A_n*(x+2)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			1;
-3,        2;
9,       -10,      3;
-23,      38,    -21,      4;
57,     -122,     99,    -36,      5;
-135,    358,   -381,    204,    -55,     6;
313,    -986,   1299,   -916,    365,   -78,     7;
-711,   2598,  -4077,   3564,  -1875,   594,  -105,    8;
1593,  -6618,  12051, -12564,   8205, -3438,   903, -136,    9;
-3527, 16422, -34029,  41196, -32115, 16722, -5817, 1304, -171, 10;

Cf. A248345, A045883, A014105, A001057.

T(n,k) = (k+1)*sum(i=0,n-k,(-2)^i*binomial(i+k+1,k+1))
for(n=0,10,for(k=0,n,print1(T(n,k),", ")))

T(n,0) = ((6*n+8)*(-2)^n+1)/9, for n >= 0.
T(n,n-1) = -n*(2*n+1), for n >= 1.
Row n sums to A001057(n+1).

cp's OEIS Frontend

A094705 Convolution of Jacobsthal(n) and 3^n.

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A124763 Number of non-rises (levels or falls) for compositions in standard order.

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A094953 Triangle T(n,m) read by rows: number of rises (drops) in the compositions of n with m parts, m>=2.

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A102301 a(n) = ((3*n + 1)*2^(n+3) + 9 + (-1)^n)/18.

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A113861 a(n) = (1/9)*((6*n - 7)*2^(n-1) - (-1)^n).

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A137241 Number triples (k,3-k,2-2k), concatenated for k=0, 1, 2, 3,...

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A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

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A102301 a(n) = ((3n + 1)2^(n+3) + 9 + (-1)^n)/18.

A113861 a(n) = (1/9)((6n - 7)*2^(n-1) - (-1)^n).

A140960 a(n) = (2(-1)^n - 2^(n+1) + 3n*2^n)/9.

A246788 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x+2)^k.