A026622 -id:A026622 - cp's OEIS frontend

A026956 Self-convolution of array T given by A026615.

1, 2, 11, 52, 200, 742, 2752, 10278, 38670, 146426, 557408, 2131318, 8179646, 31491202, 121568150, 470404274, 1823968074, 7085220834, 27567196704, 107414120214, 419080195374, 1636990646274, 6401210885934, 25055584929954, 98160790785714, 384885441746202, 1510279309724502

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A026615, A026616, A026617, A026618, A026619, A026620.
Cf. A026621, A026622, A026623, A026624, A026625, A026957, A026958.
Cf. A026959, A026960.

[n le 1 select n+1 else Catalan(n-2)*(49*n^2-105*n+48)/n - 6: n in [0..40]]; // G. C. Greubel, Jun 17 2024

Table[If[n==0, 1, CatalanNumber[n-2]*(49*n^2-105*n+48)/n -6], {n,0,40}] (* G. C. Greubel, Jun 17 2024 *)

[1,2]+[catalan_number(n-2)*(49*n^2-105*n+48)/n -6 for n in range(2,41)] # G. C. Greubel, Jun 17 2024

From G. C. Greubel, Jun 17 2024: (Start)
a(n) = Sum_{k=0..n} A026615(n, k) * A026615(n, n-k).
a(n) = A000108(n-2)*(49*n^2 - 105*n + 48)/n - 6, for n >= 1, with a(0) = 1.
G.f.: (4 - 8*x + 5*x^2 - x^3 - (3 - x + 4*x^2)*sqrt(1-4*x))/((1-x)*sqrt(1-4*x)).
E.g.f.: (1/6)*( 18 + 24*x - 36*exp(x) + 4*exp(2*x)*(6 - 6*x + x^2) * BesselI(0, 2*x) + x*exp(2*x)*(23 - 4*x)*BesselI(1, 2*x) ). (End)

More terms from Sean A. Irvine, Oct 20 2019

A026957 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A026615.

Original entry on oeis.org

1, 6, 35, 154, 613, 2362, 9028, 34510, 132241, 508210, 1958460, 7565906, 29292820, 113633930, 441579702, 1718642278, 6698377449, 26139863330, 102125977396, 399415127682, 1563614796608, 6126581578954, 24024810462810, 94281930087290, 370254213115948, 1454967778894282

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
Cf. A026622, A026623, A026624, A026625, A026956, A026958, A026959.
Cf. A026960.

[1] cat [(n-1)*Binomial(2*n,n-1)*(49*n^3 -105*n^2 +62*n -24)/( 24*Binomial(2*n,4)) -2*(2*n-1): n in [2..40]]; // G. C. Greubel, Jun 17 2024

Table[If[n==1, 1, (n-1)*Binomial[2*n,n-1]*(49*n^3 -105*n^2 +62*n -24 )/(24*Binomial[2*n,4]) - 2*(2*n-1)], {n,40}] (* G. C. Greubel, Jun 17 2024 *)

[1]+[(n-1)*binomial(2*n,n-1)*(49*n^3-105*n^2+62*n-24 )/( 24*binomial(2*n, 4)) -2*(2*n-1) for n in range(2,41)] # G. C. Greubel, Jun 17 2024

a(n) = (n-1)*binomial(2*n, n-1)*(49*n^3 - 105*n^2 + 62*n - 24 )/( 24*binomial(2*n, 4)) - 2*(2*n-1), for n >= 2, with a(1) = 1. - G. C. Greubel, Jun 17 2024

More terms from Sean A. Irvine, Oct 20 2019

A026958 a(n) = Sum_{k=0..n-2} T(n,k) * T(n,k+2), with T given by A026615.

Original entry on oeis.org

1, 10, 69, 340, 1476, 6074, 24419, 97136, 384428, 1517422, 5981070, 23556746, 92743296, 365078146, 1437124303, 5657887016, 22279053380, 87749051950, 345704345066, 1362361338578, 5370436417996, 21176724230654, 83529562154498, 329573910914930, 1300752571946396

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
Cf. A026622, A026623, A026624, A026625, A026956, A026957, A026959.
Cf. A026960.

[n eq 2 select 1 else Binomial(2*n,n+2)*(49*n^4 -154*n^3 + 209*n^2 -200*n +108)/(24*Binomial(2*n,4)) -2*(n^2-2*n+2): n in [2..40]]; // G. C. Greubel, Jun 17 2024

Table[Binomial[2*n,n+2]*(49*n^4 -154*n^3 +209*n^2 -200*n +108)/(24* Binomial[2*n,4]) -2*(n^2-2*n+2) + Boole[n==2], {n,2,40}] (* G. C. Greubel, Jun 17 2024 *)

[binomial(2*n,n+2)*(49*n^4 -154*n^3 +209*n^2 -200*n +108 )/(24*binomial(2*n,4)) -2*(n^2-2*n+2) +int(n==2) for n in range(2,41)] # G. C. Greubel, Jun 17 2024

a(n) = binomial(2*n, n+2)*(49*n^4 - 154*n^3 + 209*n^2 - 200*n + 108)/(24*binomial(2*n, 4)) -2*(n^2 - 2*n + 2) + [n=2]. - G. C. Greubel, Jun 17 2024

More terms from Sean A. Irvine, Oct 20 2019

A026959 a(n) = Sum_{k=0..n-3} T(n,k) * T(n,k+3), with T given by A026615.

Original entry on oeis.org

1, 14, 115, 640, 3049, 13494, 57491, 239768, 986976, 4027666, 16335660, 65955960, 265386251, 1064993622, 4264898875, 17051078256, 68080259516, 271537515786, 1082098938452, 4309269809044, 17151303222746, 68232856509950, 271350536990740, 1078796298028680, 4287906741748940

Offset: 3

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..1000

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
Cf. A026622, A026623, A026624, A026625, A026956, A026957, A026958.
Cf. A026960.

[n eq 3 select 1 else Binomial(2*n,n+3)*(49*n^4 -154*n^3 +279*n^2 -390*n +288)/(24* Binomial(2*n,4)) -(n-2)*(2*n^2-5*n+9)/3: n in [3..40]]; // G. C. Greubel, Jun 17 2024

Table[(2*n-4)!*(49*n^4 -154*n^3 +279*n^2 -390*n +288)/((n-3)!*(n+3)!) - (n-2)*(2*n^2-5*n+9)/3 +Boole[n==3], {n,3,40}] (* G. C. Greubel, Jun 17 2024 *)

[binomial(2*n,n+3)*(49*n^4 -154*n^3 +279*n^2 -390*n +288)/(24*binomial(2*n,4)) -(1/3)*(n-2)*(2*n^2-5*n+9) +int(n==3) for n in range(3,41)] # G. C. Greubel, Jun 17 2024

a(n) = binomial(2*n, n+3)*(49*n^4 - 154*n^3 + 279*n^2 - 390*n + 288)/(4! * binomial(2*n, 4)) - (1/3)*(n-2)*(2*n^2 - 5*n + 9) + [n=3]. - G. C. Greubel, Jun 17 2024

More terms from Sean A. Irvine, Oct 20 2019

A026960 a(n) = Sum_{k=0..n} (k+1) * A026615(n,k).

Original entry on oeis.org

1, 3, 10, 30, 78, 189, 440, 999, 2230, 4917, 10740, 23283, 50162, 107505, 229360, 487407, 1032174, 2179053, 4587500, 9633771, 20185066, 42205161, 88080360, 183500775, 381681638, 792723429, 1644167140, 3405774819, 7046430690, 14562623457, 30064771040

Offset: 0

Clark Kimberling

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

Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
Cf. A026622, A026623, A026624, A026625, A026956, A026957, A026958.
Cf. A026959.

[n le 1 select 2*n+1 else 7*(n+2)*2^(n-3) - n - 2: n in [0..40]]; // G. C. Greubel, Jun 16 2024

Join[{1,3},Table[7(n+2)2^(n-3)-n-2,{n,2,30}]] (* or *) LinearRecurrence[ {6,-13,12,-4},{1,3,10,30,78,189},30] (* Harvey P. Dale, Oct 31 2015 *)

Vec((1-3*x+5*x^2-3*x^3-4*x^4+3*x^5)/((1-x)^2*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Feb 18 2016

[7*(n+2)*2^(n-3) - n - 2 + (5/4)*int(n==0) + (3/4)*int(n==1) for n in range(41)] # G. C. Greubel, Jun 16 2024

For n>1, a(n) = 7*(n+2)*2^(n-3) - n - 2.
From Colin Barker, Feb 18 2016: (Start)
a(n) = 6*a(n-1)-13*a(n-2)+12*a(n-3)-4*a(n-4) for n>5
G.f.: (1-3*x+5*x^2-3*x^3-4*x^4+3*x^5) / ((1-x)^2*(1-2*x)^2).
(End)

A108765 Expansion of g.f. (1 - x + x^2)/((1-3x)(x-1)^2).

Original entry on oeis.org

1, 4, 14, 45, 139, 422, 1272, 3823, 11477, 34440, 103330, 310001, 930015, 2790058, 8370188, 25110579, 75331753, 225995276, 677985846, 2033957557, 6101872691, 18305618094, 54916854304, 164750562935, 494251688829, 1482755066512

Offset: 0

Creighton Dement, Jun 24 2005

Superseeker suggests a(n+2) - 2*a(n+1) + a(n) = 7*3^n = A005032(n).
Inverse binomial transform gives match with first differences of A026622.
Floretion Algebra Multiplication Program, FAMP Code: kbasefor[(- 'j + 'k - 'ii' - 'ij' - 'ik')], vesfor = A000004, Fortype: 1A, Roktype (leftfactor) is set to:Y[sqa.Findk()] = Y[sqa.Findk()] + Math.signum(Y[sqa.Findk()])*p (internal program code)

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A005032, A026622.

s=1;lst={s};Do[s+=(s+(n+=s));AppendTo[lst, s], {n, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 11 2008 *)
CoefficientList[Series[(1-x+x^2)/((1-3x)(x-1)^2),{x,0,40}],x] (* or *) LinearRecurrence[{5,-7,3},{1,4,14},40] (* Harvey P. Dale, Dec 11 2012 *)

From Rolf Pleisch, Feb 10 2008: (Start)
a(0) = 1; a(n) = 3*a(n-1) + n.
a(n) = (7*3^n - 2*n - 3)/4. (End)
a(0)=1, a(1)=4, a(2)=14, a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3). - Harvey P. Dale, Dec 11 2012

A174317 a(0)=1, a(1)=2, a(2)=1; for n>2, a(n) = 7*2^(n-3)-2.

Original entry on oeis.org

1, 2, 1, 5, 12, 26, 54, 110, 222, 446, 894, 1790, 3582, 7166, 14334, 28670, 57342, 114686, 229374, 458750, 917502, 1835006, 3670014, 7340030, 14680062, 29360126, 58720254, 117440510, 234881022, 469762046, 939524094, 1879048190, 3758096382

Offset: 0

Richard Choulet, Mar 15 2010

			a(4) = 14-2 = 12.
a(5) = 7*4-2 = 26.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A174314, A174316

taylor(1+2*z+z^2+5*z^3-((2*z^4)/(1-z))+((14*z^4)/(1-2*z)),z=0,50);

G.f: (1+x-x^2+4*x^3-7*x^4)/(1-x)+(14*x^4)/(1-2*x).

a(n) = A026622(n-1), n>2. a(n) = A176448(n-3), n>2. - R. J. Mathar, Mar 01 2016

Definition edited by Olivier Gérard, Oct 24 2012

cp's OEIS Frontend

A026956 Self-convolution of array T given by A026615.

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A026957 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A026615.

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A026958 a(n) = Sum_{k=0..n-2} T(n,k) * T(n,k+2), with T given by A026615.

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A026959 a(n) = Sum_{k=0..n-3} T(n,k) * T(n,k+3), with T given by A026615.

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A026960 a(n) = Sum_{k=0..n} (k+1) * A026615(n,k).

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A108765 Expansion of g.f. (1 - x + x^2)/((1-3*x)*(x-1)^2).

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A174317 a(0)=1, a(1)=2, a(2)=1; for n>2, a(n) = 7*2^(n-3)-2.

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A108765 Expansion of g.f. (1 - x + x^2)/((1-3x)(x-1)^2).