A077415 -id:A077415 - cp's OEIS frontend

A299120 a(n) = (n-1)(n-2)(n+3)*(n+2)/12.

1, 0, 0, 5, 21, 56, 120, 225, 385, 616, 936, 1365, 1925, 2640, 3536, 4641, 5985, 7600, 9520, 11781, 14421, 17480, 21000, 25025, 29601, 34776, 40600, 47125, 54405, 62496, 71456, 81345, 92225, 104160, 117216, 131461, 146965, 163800, 182040, 201761, 223041

Offset: 0

Juri-Stepan Gerasimov, Feb 03 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001263, A033275, A077415, A299146, A299198.

List([0..10^3], n->n^4/12+n^3/6-7*n^2/12-2*n/3+1); # Muniru A Asiru, Feb 04 2018

[n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1: n in [0..40]];

seq(n^4/12+n^3/6-7*n^2/12-2*n/3+1, n=0..10^3); # Muniru A Asiru, Feb 04 2018

Rest@ CoefficientList[Series[(1 - 5 x + 10 x^2 - 5 x^3 + x^4)/(1 - x)^5, {x, 0, 41}], x] (* Michael De Vlieger, Feb 10 2018 *)
f[n_] := n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1; Array[f, 40, 0] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 0, 0, 5, 21}, 40] (* Robert G. Wilson v, Mar 12 2018 *)

Vec((1 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5 + O(x^50)) \\ Colin Barker, Feb 05 2018

a(n) = n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1 = (n-1)*(n-2)*(n+3)*(n+2)/12.
From Colin Barker, Feb 05 2018: (Start)
G.f.: (1 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5. (End)
a(n) = A033275(n+2) for n > 1. - Georg Fischer, Oct 09 2018
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=3} 1/a(n) = 43/150.
Sum_{n>=3} (-1)^(n+1)/a(n) = 16*log(2)/5 - 154/75. (End)
E.g.f.: exp(x)*(12 - 12*x + 6*x^2 + 8*x^3 + x^4)/12. - Stefano Spezia, Feb 21 2024

Edited by Wolfdieter Lang, Apr 06 2018

A299198 a(n) = n^4/6 - 2n^3/3 - n^2/6 + 5n/3 + 1.

Original entry on oeis.org

2, 1, 0, 5, 26, 77, 176, 345, 610, 1001, 1552, 2301, 3290, 4565, 6176, 8177, 10626, 13585, 17120, 21301, 26202, 31901, 38480, 46025, 54626, 64377, 75376, 87725, 101530, 116901, 133952, 152801, 173570, 196385, 221376, 248677, 278426, 310765, 345840, 383801, 424802, 469001

Offset: 1

Juri-Stepan Gerasimov, Feb 04 2018

			For n=2, a(2) = 1^4/6 - 2*1^3/3 - 1^2/6 + 5*1/3 + 1 = 2.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001263, A067998, A077415, A299120, A299146.

List([1..50], n -> n^4/6-2*n^3/3-n^2/6+5*n/3+1); # Muniru A Asiru, Feb 04 2018

[div((n-3)*(n+1)*(n^2-2*n-2),6) for n in 1:50] |> println # Bruno Berselli, Apr 11 2018

[n^4/6-2*n^3/3-n^2/6+5*n/3+1: n in [1..50]];

seq(n^4/6-2*n^3/3-n^2/6+5*n/3+1,n=1..50); # Muniru A Asiru, Feb 04 2018

f[n_] := n^4/6 - 2 n^3/3 - n^2/6 + 5 n/3 + 1; Array[f, 50] (* or *)
CoefficientList[ Series[(-2 + 9 x - 15 x^2 + 5 x^3 - x^4)/(-1 + x)^5, {x, 0, 50}], x] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {2, 1, 0, 5, 26}, 50] (* Robert G. Wilson v, Feb 09 2018 *)

Vec(x*(2 - 9*x + 15*x^2 - 5*x^3 + x^4) / (1 - x)^5 + O(x^50)) \\ Colin Barker, Feb 05 2018

a(n) = (n - 3)*(n + 1)*(n^2 - 2*n - 2)/6 = A299120(n-1) + A299120(1-n).
From Colin Barker, Feb 05 2018: (Start)
G.f.: x*(2 - 9*x + 15*x^2 - 5*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)
E.g.f.: exp(x)*(6 + 6*x - 6*x^2 + 2*x^3 + x^4)/6. - Iain Fox, Feb 09 2018
6*a(n) = A067998(n)^2 - 5*A067998(n) + 6. - Bruno Berselli, Apr 11 2018

A092081 Triangle of certain double factorials.

Original entry on oeis.org

1, 1, 2, 1, 3, 8, 1, 4, 15, 48, 1, 5, 24, 105, 384, 1, 6, 35, 192, 945, 3840, 1, 7, 48, 315, 1920, 10395, 46080, 1, 8, 63, 480, 3465, 23040, 135135, 645120, 1, 9, 80, 693, 5760, 45045, 322560, 2027025, 10321920, 1, 10, 99, 960, 9009, 80640, 675675, 5160960

Offset: 0

Wolfdieter Lang, Mar 19 2004

This is the rectangular array A(3;m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0, read by SW-NE diagonals. For n!! see A006882 (double factorials).

W. Lang, First 9 rows.

Diagonals give: A000165 (double factorials of 2*n), A001147(n+1), A002866, A051577-83.

Columns give: A000012 (powers of 1), A000027 (naturals >=2), A005563, 3*A077415, for n=0..3.

a(m, n)=(n+m)!!/(m-n)!!, 0<=n<=m, else 0, with 0!! := 1.

A145069 a(n) = n(n^2 + 3n + 5)/3.

Original entry on oeis.org

0, 3, 10, 23, 44, 75, 118, 175, 248, 339, 450, 583, 740, 923, 1134, 1375, 1648, 1955, 2298, 2679, 3100, 3563, 4070, 4623, 5224, 5875, 6578, 7335, 8148, 9019, 9950, 10943, 12000, 13123, 14314, 15575, 16908, 18315, 19798, 21359, 23000, 24723, 26530

Offset: 0

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

Old name was: Partial sums of A002061, starting at n=2.

Number of floating point dot operations (multiplications and divisions) in the factorization of an (n+1) X (n+1) real matrix by Gaussian elimination as, e.g., implemented in LINPACK subroutines sgefa.f or dgefa.f. The number of multiplications alone is given by A007290. The number of additions is given by A000330. - Hugo Pfoertner, Mar 28 2018

			a(2) = a(1) + 2^2 + 2 + 1 = 3 + 4 + 2 + 1 = 10.
a(3) = a(2) + 3^2 + 3 + 1 = 10 + 9 + 3 + 1 = 23.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (corrected by Ray Chandler, Jan 19 2019)
Cleve Moler, LINPACK subroutine sgefa.f, University of New Mexico, Argonne National Lab, 1978.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002061 (n^2 - n + 1).
Cf. A028387 (n + (n+1)^2).
Cf. A077415 (zero followed by partial sums of A028387, starting at n=1).
Cf. A007290.

I:=[0, 3, 10, 23]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 30 2012

A145069:=n->n*(n^2+3*n+5)/3: seq(A145069(n), n=0..100); # Wesley Ivan Hurt, Aug 21 2014

lst={};s=0;Do[s+=n^2+n+1;AppendTo[lst,s-1],{n,0,5!}];lst
CoefficientList[Series[x(3-2*x+x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 30 2012 *)
Table[n (n^2+3n+5)/3,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,3,10,23},50] (* Harvey P. Dale, Sep 10 2016 *)

{a=0; for(n=1, 42, print1(a, ", "); a=a+n^2+n+1)} \\ adapted by Michel Marcus, Aug 23 2014

G.f.: x*(3-2*x+x^2)/(1-x)^4.
a(n) = Sum_{j=2..n+1} A002061(j).
a(n) = a(n-1) + n^2 + n + 1 for n > 0, with a(0) = 0.
a(n) = n*(n^2+3*n+5)/3. - Bruno Berselli, Apr 01 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 30 2012
a(n) = Sum_{i=1..n} 3i+(n-i)^2. - Wesley Ivan Hurt, Aug 21 2014
a(n) = A007290(n+2) + n. - Hugo Pfoertner, Mar 28 2018

Edited by Klaus Brockhaus, Oct 21 2008
G.f. adapted to the offset by Bruno Berselli, Apr 01 2011
Name, offset, and formulas changed by Wesley Ivan Hurt, Aug 21 2014

A154232 a(2n) = (n^2-n-1) + a(2n-2), a(2n+1) = (n^2-n-1)*a(2n-1), with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, -1, -1, 0, -1, 5, -5, 16, -55, 35, -1045, 64, -30305, 105, -1242505, 160, -68337775, 231, -4851982025, 320, -431826400225, 429, -47069077624525, 560, -6166049168812775, 715, -955737621165980125, 896, -172988509431042402625

Offset: 0

Roger L. Bagula, Jan 05 2009

sign

Essentially A077415 and A130031 interleaved, see formulas.

G. C. Greubel, Table of n, a(n) for n = 0..351

Cf. A077415, A130031.

function a(n)
  if n lt 2 then return n;
  elif (n mod 2 eq 0) then return ((n^2-2*n-4)/4) + a(n-2);
  else return ((n^2-4*n-1)/4)*a(n-2);
  end if; return a;
end function;
[a(n): n in [0..40]]; // G. C. Greubel, Mar 02 2021

a[0]:= 0: a[1]:= 1:
for n from 1 to 49 do
  a[2*n]:= (n^2-n-1) +a[2*n-2];
  a[2*n+1]:= (n^2-n-1)*a[2*n-1];
od:
seq(a[i],i=0..99); # Robert Israel, Sep 06 2016

(* First program *)
b[n_]:= b[n]= If[n==0, 0, n^2 -n -1 + b[n-1]];
c[n_]:= c[n]= If[n==0, 1, (n^2 -n -1)*c[n-1]];
Flatten[Table[{b[n], c[n]}, {n, 0, 15}]] (* modified by G. C. Greubel, Mar 02 2021 *)
(* Second program *)
a[n_]:= a[n]= If[n<2, n, If[EvenQ[n], ((n^2-2*n-4)/4) + a[n-2], ((n^2-4*n-1)/4)*a[n-2]]];
Table[a[n], {n,0,40}] (* G. C. Greubel, Mar 02 2021 *)

def a(n):
    if (n<2): return n
    elif (n%2==0): return ((n^2-2*n-4)/4) + a(n-2)
    else: return ((n^2-4*n-1)/4)*a(n-2)
[a(n) for n in (0..40)] # G. C. Greubel, Mar 02 2021

From Robert Israel, Sep 06 2016: (Start)
a(2*n) = A077415(n) for n >= 2.
a(2*n+1) = cos(Pi*sqrt(5)/2)*Gamma(n+1/2-sqrt(5)/2)*Gamma(n+1/2+sqrt(5)/2)/Pi.
a(2*n+1) = (-1)^n*A130031(n). (End)

Edited by Robert Israel, Sep 06 2016

A229834 Expansion of (1+4x+x^2) / ((1-x)^3(1+x)^4).

Original entry on oeis.org

1, 3, 1, 11, -2, 26, -10, 50, -25, 85, -49, 133, -84, 196, -132, 276, -195, 375, -275, 495, -374, 638, -494, 806, -637, 1001, -805, 1225, -1000, 1480, -1224, 1768, -1479, 2091, -1767, 2451, -2090, 2850, -2450, 3290, -2849, 3773, -3289, 4301, -3772, 4876, -4300, 5500, -4875, 6175, -5499, 6903, -6174, 7686, -6902

Offset: 0

Stefano Maruelli, Dec 19 2013

The sequence can be generated in the following way:
---------------------------       --------------------------
      [0]  [1] [2] [3]  [4]  ...             [i]
---------------------------       --------------------------
[0]    1,   1,  1,  1,   1,  ...  t(0,i) =    1
[1]    7,   6,  5,  4,   3,  ...  t(1,i) = t(1,i-1) - t(0,i)
[2]   19,  13,  8,  4,   1,  ...  t(2,i) = t(2,i-1) - t(1,i)
[3]   37,  24, 16, 12,  11,  ...  t(3,i) = t(3,i-1) - t(2,i)
[4]   61,  37, 21,  9,  -2,  ...  t(4,i) = t(4,i-1) - t(3,i)
[5]   91,  54, 33, 24,  26,  ...             etc.
[6]  127,  73, 40, 16, -10,  ...
[7]  169,  96, 56, 40,  50,  ...
[8]  217, 121, 65, 25, -25,  ...
[9]  271, 150, 85, 60,  85,  ...
...
Column 0 is A003215;
column 1 is A032528;
column 2 is A001082;
column 3 is A241496;
column 4 is this sequence.
The third differences are 16, -35, 64, -105, 160, ..., a signed variant of A077415. - R. J. Mathar, Apr 18 2014

Index entries for linear recurrences with constant coefficients, signature (-1,3,3,-3,-3,1,1).

Cf. A077415; A058373: a(2k) = -A058373(k); A051925: a(2k+1) = A051925(k+2).

Columns of the table in Comments section: A001082, A003215, A032528.

Table[1 + n (n + 5) (9 - (2 n + 5) (-1)^n)/48, {n, 0, 60}] (* Bruno Berselli, Apr 22 2014 *)
CoefficientList[Series[(1+4x+x^2)/((1-x)^3(1+x)^4),{x,0,60}],x] (* or *) LinearRecurrence[{-1,3,3,-3,-3,1,1},{1,3,1,11,-2,26,-10},60] (* Harvey P. Dale, Jan 27 2022 *)

G.f.: (1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^4). - R. J. Mathar, Apr 18 2014

a(n) = a(-n-5) = 1 + n*(n + 5)*(9 - (2*n + 5)*(-1)^n)/48. [Bruno Berselli, Apr 22 2014]

A254443 Numbers n such that T(n) + T(n+1) + ... + T(n+21) is a square, where T(m) = A000217(m) is the m-th triangular number.

Original entry on oeis.org

35, 75, 911, 1707, 18383, 34263, 366947, 683751, 7320755, 13640955, 146048351, 272135547, 2913646463, 5429070183, 58126881107, 108309268311, 1159623975875, 2160756296235, 23134352636591, 43106816656587, 461527428756143, 859975576835703, 9207414222486467

Offset: 1

Colin Barker, May 04 2015

Positive integers y in the solutions to 2*x^2-22*y^2-484*y-3542 = 0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,20,-20,-1,1).

Cf. A176541, A176542, A000217, A000292, A001110, A077415.

Cf. A116476 (length 11), A257293 (length 13).

Vec(x*(9*x^4+4*x^3-136*x^2-40*x-35)/((x-1)*(x^4-20*x^2+1)) + O(x^100))

G.f.: x*(9*x^4+4*x^3-136*x^2-40*x-35) / ((x-1)*(x^4-20*x^2+1)).

A370912 a(n) = n(n + 2)(n + 4).

Original entry on oeis.org

0, 15, 48, 105, 192, 315, 480, 693, 960, 1287, 1680, 2145, 2688, 3315, 4032, 4845, 5760, 6783, 7920, 9177, 10560, 12075, 13728, 15525, 17472, 19575, 21840, 24273, 26880, 29667, 32640, 35805, 39168, 42735, 46512, 50505, 54720, 59163, 63840, 68757, 73920

Offset: 0

Peter Luschny, Mar 05 2024

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cases of A370419(n, k): A000012 (n=0), A001477 (n=1), A005563 (n=2), this sequence (n=3), A190577(n=4).
Cf. A077415, A370890.
Cf. A028347, A028387, A211441.

a := n -> n*(n + 2)*(n + 4): seq(a(n), n = 0..40);
# Using the generating function:
gf := 3*x*(x^2 - 4*x + 5)/(x - 1)^4: ser := series(gf, x, 42):
seq(coeff(ser, x, n), n = 0..40);

Table[n(n+2)(n+4), {n,0,40}] (* or *) CoefficientList[Series[3*x*(x^2 - 4*x + 5)/(x - 1)^4,{x,0,40}],x] (* James C. McMahon, Mar 05 2024 *)

a(n) = 8*Pochhammer(n/2, 3).
a(n) = [x^n] 3*x*(x^2 - 4*x + 5)/(x - 1)^4.
a(n) = 3 * A077415(n + 2).
From Klaus Purath, Aug 02 2024: (Start)
a(n)^2 = A028347(n+2)^3 + 4*A028347(n+2)^2.
a(n+1) - a(n) = A211441(n+2).
a(n) = 3*Sum_{i = 1..n} A028387(i). (End)
E.g.f.: exp(x)*x*(15 + 9*x + x^2). - Stefano Spezia, Aug 18 2024
From Amiram Eldar, Oct 03 2024: (Start)
Sum_{n>=1} 1/a(n) = 11/96.
Sum_{n>=1} (-1)^(n+1)/a(n) = 5/96. (End)

cp's OEIS Frontend

A299120 a(n) = (n-1)*(n-2)*(n+3)*(n+2)/12.

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A299198 a(n) = n^4/6 - 2*n^3/3 - n^2/6 + 5*n/3 + 1.

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A092081 Triangle of certain double factorials.

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A145069 a(n) = n*(n^2 + 3*n + 5)/3.

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Magma

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A154232 a(2n) = (n^2-n-1) + a(2n-2), a(2n+1) = (n^2-n-1)*a(2n-1), with a(0)=0 and a(1)=1.

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A229834 Expansion of (1+4*x+x^2) / ((1-x)^3*(1+x)^4).

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Mathematica

Formula

A254443 Numbers n such that T(n) + T(n+1) + ... + T(n+21) is a square, where T(m) = A000217(m) is the m-th triangular number.

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A299120 a(n) = (n-1)(n-2)(n+3)*(n+2)/12.

A299198 a(n) = n^4/6 - 2n^3/3 - n^2/6 + 5n/3 + 1.

A145069 a(n) = n(n^2 + 3n + 5)/3.

A229834 Expansion of (1+4x+x^2) / ((1-x)^3(1+x)^4).

A370912 a(n) = n(n + 2)(n + 4).