A045618 -id:A045618 - cp's OEIS frontend

A154222 Row sums of number triangle A154221.

1, 2, 4, 8, 17, 38, 87, 200, 457, 1034, 2315, 5132, 11277, 24590, 53263, 114704, 245777, 524306, 1114131, 2359316, 4980757, 10485782, 22020119, 46137368, 96469017, 201326618, 419430427, 872415260, 1811939357, 3758096414, 7784628255, 16106127392, 33285996577

Offset: 0

Paul Barry, Jan 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A045618, A154221.

[(1/4)*(4*(n+1)+(n-1)*2^n+0^n): n in [0..35]]; // Vincenzo Librandi, Sep 07 2016

Join[{1},LinearRecurrence[{6, -13, 12, -4}, {2, 4, 8,17}, 25]] (* or *) Table[(1/4)*( 4*(n+1) + (n-1)*2^n + 0^n), {n,0,25}] (* G. C. Greubel, Sep 06 2016 *)

Vec((x^4-2*x^3+5*x^2-4*x+1)/((x-1)^2*(2*x-1)^2) + O(x^100)) \\ Colin Barker, Oct 11 2014

a(n) = (1/4)*( 4*(n+1) + (n-1)*2^n + 0^n).
From Colin Barker, Oct 11 2014: (Start)
a(n) = A045618(n-4) + 2^n for n>3.
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4) for n>4.
a(n) = (4 - 2^n + (4+2^n)*n)/4 for n>0.
G.f.: (x^4 - 2*x^3 + 5*x^2 - 4*x + 1) / ((x-1)^2*(2*x-1)^2).
(End)
E.g.f.: (1/4)*(1 + 4*(1 + x)*exp(x) + (2*x - 1)*exp(2*x)). - G. C. Greubel, Sep 06 2016

More terms and xrefs from Colin Barker, Oct 11 2014

A206306 Riordan array (1, x/(1-3x+2x^2)).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 7, 6, 1, 0, 15, 23, 9, 1, 0, 31, 72, 48, 12, 1, 0, 63, 201, 198, 82, 15, 1, 0, 127, 522, 699, 420, 125, 18, 1, 0, 255, 1291, 2223, 1795, 765, 177, 21, 1, 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1

Offset: 0

Philippe Deléham, Feb 06 2012

The convolution triangle of the Mersenne numbers A000225. - Peter Luschny, Oct 09 2022

			Triangle begins:
  1;
  0,    1;
  0,    3,    1;
  0,    7,    6,     1;
  0,   15,   23,     9,     1;
  0,   31,   72,    48,    12,     1;
  0,   63,  201,   198,    82,    15,    1;
  0,  127,  522,   699,   420,   125,   18,    1;
  0,  255, 1291,  2223,  1795,   765,  177,   21,   1;
  0,  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;
  0, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27,  1;

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

Columns: A000007, A000225 (Mersenne numbers), A045618, A055582.
Cf. A008585, A009545, A084938, A062725, A110441, A204089, A204091.
Cf. A045741, A078020, A244038.

function T(n,k) // T = A206306
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  elif k eq 0 then return 0;
  else return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 20 2022

# Uses function PMatrix from A357368.
PMatrix(10, n -> 2^n - 1); # Peter Luschny, Oct 09 2022

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==0, 0, 3*T[n- 1, k] +T[n-1, k-1] -2*T[n-2, k]]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 20 2022 *)

def T(n,k): # T = A206306
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==0): return 0
    else: return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 20 2022

Triangle T(n,k), read by rows, given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Diagonals sums are even-indexed Fibonacci numbers.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A204089(n), A204091(n) for x = 0, 1, 2 respectively.
G.f.: (1-3*x+2*x^)/(1-(3+y)*x+2*x^2).
From Philippe Deléham, Nov 17 2013; corrected Feb 13 2020: (Start)
T(n, n) = 1.
T(n+1, n) = 3n = A008585(n).
T(n+2, n) = A062725(n).
T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), T(0,0)=T(1,1)=T(2,2)=1, T(1,0)=T(2,0)=0, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. (End)
From G. C. Greubel, Dec 20 2022: (Start)
Sum_{k=0..n} (-1)^k*T(n,k) = [n=1] - A009545(n).
Sum_{k=0..n} (-2)^k*T(n,k) = [n=1] + A078020(n+1).
T(2*n, n+1) = A045741(n+2), n >= 0.
T(2*n+1, n+1) = A244038(n). (End)

A279230 Expansion of 1/((1-x)^2(1-2x+2*x^2)).

Original entry on oeis.org

1, 4, 9, 14, 15, 8, -7, -22, -21, 12, 77, 142, 143, 16, -239, -494, -493, 20, 1045, 2070, 2071, 24, -4071, -8166, -8165, 28, 16413, 32798, 32799, 32, -65503, -131038, -131037, 36, 262181, 524326, 524327, 40, -1048535, -2097110, -2097109, 44, 4194349, 8388654, 8388655

Offset: 0

Philippe Deléham, Dec 08 2016

Partial sums of A077860.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-2).

Cf. A001477, A045618, A077859, A077860, A279231.

Vec(1 / ((1 - x)^2*(1 - 2*x + 2*x^2)) + O(x^50)) \\ Colin Barker, Aug 04 2017

{a(n) = sum(k=0, n\2, (-1)^k*binomial(n+3, 2*k+3))} \\ Seiichi Manyama, Apr 07 2019

a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 2*a(n-4) for n>3.
a(n) = 2*a(n-1) - 2*a(n-2) + n + 1, with a(-1) = a(-2) = 0.
a(n) = (3 - (1-i)^(1+n) - (1+i)^(1+n) + n) where i=sqrt(-1). - Colin Barker, Aug 04 2017
From Seiichi Manyama, Apr 07 2019: (Start)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n+3,2*k+3).
a(n) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+1,j+1) * binomial(n-i+1,j+1). (End)

A279231 Expansion of g.f. 1/((1 - x)^2(1 - 3x + 3*x^2)).

Original entry on oeis.org

1, 5, 15, 34, 62, 90, 91, 11, -231, -716, -1444, -2172, -2171, 17, 6579, 19702, 39386, 59070, 59071, 23, -177123, -531416, -1062856, -1594296, -1594295, 29, 4782999, 14348938, 28697846, 43046754, 43046755, 35, -129140127, -387420452, -774840940, -1162261428

Offset: 0

Philippe Deléham, Dec 08 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,9,-3).

Cf. A001477, A045618, A077859.

CoefficientList[Series[1/((1-x)^2(1-3x+3x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{5,-10,9,-3},{1,5,15,34},40] (* Harvey P. Dale, Aug 03 2025 *)

Vec(1/((1-x)^2*(1-3*x+3*x^2)) + O(x^30)) \\ Colin Barker, Dec 08 2016

a(n) = 5*a(n-1) - 10*a(n-2) + 9*a(n-3) - 3*a(n-4) for n > 3.
a(n) = 3*a(n-1) - 3*a(n-2) + n + 1, with a(-1) = a(-2) = 0.
a(n) = 4 + n + 3^(1+n/2)*(sqrt(3)*sin(n*Pi/6) - cos(n*Pi/6)). - Stefano Spezia, Feb 11 2023
a(n) = Sum_{k=0..floor(n/3)} (-1)^k*binomial(n+4,3*k+4). - Taras Goy, Jan 03 2025
E.g.f.: exp(x)*(4 + x - 3*exp(x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))). - Stefano Spezia, Jan 03 2025

Incorrect term corrected by Colin Barker, Dec 09 2016

A320998 Number of pseudo-square convex polyominoes with semiperimeter n.

Original entry on oeis.org

1, 12, 44, 142, 399, 1044, 2571, 6168, 14357, 32786, 73746, 163872, 360462, 786468, 1703949, 3670040, 7864353, 16777260, 35651579, 75497508, 159383591, 335544350, 704643087, 1476395064, 3087007733, 6442451004, 13421772816, 27917287460, 57982058547, 120259084318

Offset: 6

N. J. A. Sloane, Oct 30 2018

nonn

The offset is not specified but appears to be 6.

Andrew Howroyd, Table of n, a(n) for n = 6..1000
Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol. 13, (2006). See Section 4.2.

Cf. A045618, A320999.

seq(coeff(series(2*x^6/((1-x)^2*(1-2*x)^2)-add(k*x^(3*(k+1))/(1-x^(k+1))^2,k=1..ceil(n/3)),x,n+1), x, n), n = 6 .. 35); # Muniru A Asiru, Oct 31 2018

seq[n_] := 2*x^6/((1 - x)^2*(1 - 2*x)^2) - Sum[k*x^(3*(k + 1))/(1 - x^(k + 1))^2 + O[x]^(6 + n), {k, 1, Ceiling[n/3]}] // CoefficientList[#, x]& // Drop[#, 6]&;
seq[30] (* Jean-François Alcover, Sep 07 2019, from PARI *)

seq(n)={Vec(2*x^6/((1-x)^2*(1-2*x)^2) - sum(k=1, ceil(n/3), k*x^(3*(k+1))/(1-x^(k+1))^2 + O(x^(6+n))))} \\ Andrew Howroyd, Oct 31 2018

a(n) = 2*A045618(6+n) - A320999(n). - Andrew Howroyd, Oct 31 2018

Terms a(16) and beyond from Andrew Howroyd, Oct 31 2018

A106194 Triangle read by rows, generated from binomial transforms of odd numbers.

Original entry on oeis.org

1, 4, 1, 12, 5, 1, 32, 17, 6, 1, 80, 49, 23, 7, 1, 192, 129, 72, 30, 8, 1, 448, 321, 201, 102, 38, 9, 1, 1024, 769, 522, 303, 140, 47, 10, 1, 2304, 1793, 1291, 825, 443, 187, 57, 11, 1, 5120, 4097, 3084, 2116, 1268, 630, 244, 68, 12, 1

Offset: 0

Gary W. Adamson, Apr 24 2005

Appending the binomial transform of the natural numbers, (A001792: 1, 3, 8, 20, 48...) to A106194 as a leftmost column creates triangle A055249.
Placing zeros into the offset spaces, column 1: 0, 1, 5, 17, 49...; is the binomial transform of 0, 1, 3, 5...; and alternatively the binomial transform of 0, 0, 1, 2, 3...
n-th column is the binomial transform of 1, 3, 5...prefaced by n zeros. n-th column is alternatively the binomial transform of 1, 2, 3...prefaced by (n+1) zeros. The triangle of A106194 is identical to the binomial transform (of natural numbers, prefaced with zeros) triangle: A055249, deleting the leftmost column.

			First few rows of the triangle are:
1;
4, 1;
12, 5, 1;
32, 17, 6, 1;
80, 49, 23, 7, 1;
192, 129, 72, 30, 8, 1;
448, 321, 201, 102, 38, 9, 1;
...

Cf. A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055249.

A124759 Sum of products of consecutive terms for compositions in standard order.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 2, 0, 3, 4, 3, 3, 4, 3, 3, 0, 4, 6, 4, 6, 6, 4, 4, 4, 6, 6, 5, 4, 5, 4, 4, 0, 5, 8, 5, 9, 8, 5, 5, 8, 9, 8, 7, 5, 6, 5, 5, 5, 8, 9, 7, 8, 8, 6, 6, 5, 7, 7, 6, 5, 6, 5, 5, 0, 6, 10, 6, 12, 10, 6, 6, 12, 12, 10, 9, 6, 7, 6, 6, 10, 12, 12, 10, 10, 10, 8, 8, 6, 8, 8, 7, 6, 7, 6, 6, 6, 10

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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

Alois P. Heinz, Rows n = 0..14, flattened

Cf. A066099, A070939, A011782 (row lengths), A045618 (row sums).

For a composition b(1),...,b(k), a(n) = Sum_{i=1}^{k-1} b(i) b(i+1).

A159755 Triangle of C. A. Laisant for k<=0 (see A062111 and A152920).

Original entry on oeis.org

0, -1, 1, -2, -1, 4, -3, -3, 0, 12, -4, -5, -4, 4, 32, -5, -7, -8, -4, 16, 80, -6, -9, -12, -12, 0, 48, 192, -7, -11, -16, -20, -16, 16, 128, 448, -8, -13, -20, -28, -32, -16, 64, 320, 1024, -9, -15, -24, -36, -48, -48, 0, 192, 768, 2304

Offset: 0

Philippe Deléham, Apr 21 2009

			Triangle begins : 0 ; -1,1 ; -2,-1,4 ; -3,-3,0,12 ; -4,-5,-4,4,32 ; ...

Sum_{k=0..n}T(n,k)= A045618(n-2)for n>=2 . T(2n,n)=-A001787(n).

A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).

Original entry on oeis.org

1, 1, -4, -32, 400, 6912, -153664, -4194304, 136048896, 5120000000, -219503494144, -10567230160896, 564668382613504, 33174037869887488, -2125764000000000000, -147573952589676412928, 11034809241396899282944, 884295678882933431599104, -75613185918270483380568064

Offset: 1

Rigoberto Florez, Aug 26 2018

sign

Discriminant of Fibonacci polynomials.

Fibonacci polynomials are defined as F(0)=0, F(1)=1 and F(n)=x*F(n-1)+F(n-2) for n>1. Coefficients are given in triangle A168561 with offset 1.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, 2018.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A168561, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

Essentially the same as A127670.

[(-1)^((n-2)*(n-1) div 2)*2^(n-1)*n^(n-3): n in [1..20]]; // Vincenzo Librandi, Aug 27 2018

Array[(-1)^((#-2)*(#-1)/2)*2^(#-1)*#^(#-3)&,20]

concat([1], [poldisc(p) | p<-Vec(x/(1-x^2-y*x) - x + O(x^20))]) \\ Andrew Howroyd, Aug 26 2018

A383841 Expansion of 1/((1-x) * (1-2x) (1-3*x))^2.

Original entry on oeis.org

1, 12, 86, 480, 2307, 10044, 40792, 157440, 584693, 2107596, 7420218, 25634880, 87207559, 292924668, 973531964, 3206704800, 10482373305, 34042285260, 109930177630, 353238247200, 1130137576331, 3601849005372, 11440208166816, 36225346150080, 114391746903037, 360325587293004

Offset: 0

Seiichi Manyama, May 12 2025

Index entries for linear recurrences with constant coefficients, signature (12,-58,144,-193,132,-36).

Column k=3 of A383843.

Cf. A000392, A045618.

a(n) = sum(k=0, n, stirling(k+3, 3, 2)*stirling(n-k+3, 3, 2));

a(n) = 12*a(n-1) - 58*a(n-2) + 144*a(n-3) - 193*a(n-4) + 132*a(n-5) - 36*a(n-6).

a(n) = Sum_{k=0..n} Stirling2(k+3,3) * Stirling2(n-k+3,3).

cp's OEIS Frontend

A154222 Row sums of number triangle A154221.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A206306 Riordan array (1, x/(1-3*x+2*x^2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A279230 Expansion of 1/((1-x)^2*(1-2*x+2*x^2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A279231 Expansion of g.f. 1/((1 - x)^2*(1 - 3*x + 3*x^2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A320998 Number of pseudo-square convex polyominoes with semiperimeter n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A106194 Triangle read by rows, generated from binomial transforms of odd numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

A124759 Sum of products of consecutive terms for compositions in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A159755 Triangle of C. A. Laisant for k<=0 (see A062111 and A152920).

Views

A206306 Riordan array (1, x/(1-3x+2x^2)).

A279230 Expansion of 1/((1-x)^2(1-2x+2*x^2)).

A279231 Expansion of g.f. 1/((1 - x)^2(1 - 3x + 3*x^2)).

A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).

A383841 Expansion of 1/((1-x) * (1-2x) (1-3*x))^2.