A036827 -id:A036827 - cp's OEIS frontend

A036828 A036827/2.

0, 1, 17, 125, 637, 2637, 9549, 31501, 97037, 283661, 795661, 2158605, 5697549, 14696461, 37175309, 92471309, 226689037, 548667405, 1313079309, 3111125005, 7305429005, 17016291341, 39346765837, 90378862605, 206342979597, 468486979597, 1058239676429

Offset: 0

N. J. A. Sloane

This sequence is related to A036826 by a(n) = n*A036826(n) - Sum_{i=0..n-1} A036826(i). - Bruno Berselli, Mar 06 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-32,56,-48,16).

Cf. A036826, A209359.

m:=26; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(-(4*x^2+8*x+1)/((x-1)*(2*x-1)^4))); // Bruno Berselli, Mar 06 2012

LinearRecurrence[{9, -32, 56, -48, 16}, {0, 1, 17, 125, 637}, 27] (* Bruno Berselli, Mar 06 2012 *)

a(n) = 2^n*(n^3-3*n^2+9*n-13)+13 \\ Bruno Berselli, Mar 06 2012

G.f.: -x*(4*x^2+8*x+1)/((x-1)*(2*x-1)^4). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 13 2009

a(n) = 2^n*(n^3-3*n^2+9*n-13)+13. - Bruno Berselli, Mar 06 2012

Typo in definition corrected by R. J. Mathar, Sep 16 2009

A130472 A permutation of the integers: a(n) = (-1)^n * floor( (n+1)/2 ).

Original entry on oeis.org

0, -1, 1, -2, 2, -3, 3, -4, 4, -5, 5, -6, 6, -7, 7, -8, 8, -9, 9, -10, 10, -11, 11, -12, 12, -13, 13, -14, 14, -15, 15, -16, 16, -17, 17, -18, 18, -19, 19, -20, 20, -21, 21, -22, 22, -23, 23, -24, 24, -25, 25, -26, 26, -27, 27, -28, 28, -29, 29, -30, 30, -31, 31, -32, 32

Offset: 0

Clark Kimberling, May 28 2007

Pisano period lengths: 1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, ... - R. J. Mathar, Aug 10 2012

Partial sums of A038608. - Stanislav Sykora, Nov 27 2013

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (-1,1,1).
Index entries for sequences that are permutations of the integers

Sums of the form Sum_{k=0..n} k^p * q^k: A059841 (p=0,q=-1), this sequence (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

Cf. A001057, A038608.

[((-1)^n*(2*n+1)-1)/4: n in [0..80]]; // Vincenzo Librandi, Mar 29 2015

A130472:=n->ceil(n/2)*(-1)^n; seq(A130472(k), k=0..50); # Wesley Ivan Hurt, Oct 22 2013

CoefficientList[Series[(x^2-x)/(1-x^2)^2, {x, 0, 64}], x] (* Geoffrey Critzer, Sep 29 2013 *)
LinearRecurrence[{-1,1,1},{0,-1,1},70] (* Harvey P. Dale, Mar 02 2018 *)

a(n)=(-1)^n*n\2 \\ Charles R Greathouse IV, Sep 02 2015

[((-1)^n*(2*n+1) - 1)/4 for n in (0..70)] # G. C. Greubel, Mar 31 2021

a(n) = -A001057(n).
a(2n) = n, a(2n+1) = -(n+1).
a(n) = Sum_{k=0..n} k*(-1)^k.
a(n) = -a(n-1) +a(n-2) +a(n-3).
G.f.: -x/( (1-x)*(1+x)^2 ). - R. J. Mathar, Feb 20 2011
a(n) = floor( (n/2)*(-1)^n ). - Wesley Ivan Hurt, Jun 14 2013
a(n) = ceiling( n/2 )*(-1)^n. - Wesley Ivan Hurt, Oct 22 2013
a(n) = ((-1)^n*(2*n+1) - 1)/4. - Adriano Caroli, Mar 28 2015
E.g.f.: (1/4)*(-exp(x) + (1-2*x)*exp(-x) ). - G. C. Greubel, Mar 31 2021

A232599 Alternating sum of cubes, i.e., Sum_{k=0..n} k^p*q^k for p=3, q=-1.

Original entry on oeis.org

0, -1, 7, -20, 44, -81, 135, -208, 304, -425, 575, -756, 972, -1225, 1519, -1856, 2240, -2673, 3159, -3700, 4300, -4961, 5687, -6480, 7344, -8281, 9295, -10388, 11564, -12825, 14175, -15616, 17152, -18785, 20519

Offset: 0

Stanislav Sykora, Nov 26 2013

			a(3) = 0^3 - 1^3 + 2^3 - 3^3 = -20.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A000578 (cubes), A011934 (absolute values), A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

[(1 - (-1)^n*(1 -6*n^2 -4*n^3))/8: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232599:= n-> (1 -(-1)^n*(1 -6*n^2 -4*n^3))/8; seq(A232599(n), n=0..30); # G. C. Greubel, Mar 31 2021

Accumulate[Times@@@Partition[Riffle[Range[0,40]^3,{1,-1},{2,-1,2}],2]] (* Harvey P. Dale, Jul 22 2016 *)

S3M1(n)=((-1)^n*(4*n^3+6*n^2-1)+1)/8;
v = vector(10001);for(k=1,#v,v[k]=S3M1(k-1))

[(1 - (-1)^n*(1 -6*n^2 -4*n^3))/8 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = ((-1)^n*(4*n^3+6*n^2-1) +1)/8.
G.f.: (-x)*(1-4*x+x^2) / ( (1-x)*(1+x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (exp(x) - (1 +10*x -18*x^2 +4*x^3)*exp(-x))/8. - G. C. Greubel, Mar 31 2021
a(n) = - 3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5). - Wesley Ivan Hurt, Mar 31 2021

A089594 Alternating sum of squares to n.

Original entry on oeis.org

-1, 3, -6, 10, -15, 21, -28, 36, -45, 55, -66, 78, -91, 105, -120, 136, -153, 171, -190, 210, -231, 253, -276, 300, -325, 351, -378, 406, -435, 465, -496, 528, -561, 595, -630, 666, -703, 741, -780, 820, -861, 903, -946, 990, -1035, 1081, -1128, 1176, -1225, 1275

Offset: 1

Jon Perry, Dec 30 2003

Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=3, a(n-1)=(-1)^(n-1)*coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 24 2010

Also triangular numbers with alternating signs. - Stanislav Sykora, Nov 26 2013

			a(6) = 1 + 4 - 9 + 16 - 25 + 36 = 3 + 7 + 11 = 21.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), this sequence (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).
Cf. A000217.
Cf. A225144. [Bruno Berselli, Jun 06 2013]

[(-1)^n*n*(n+1)/2: n in [1..50]]; // Vincenzo Librandi, Nov 16 2011

seq(sum(binomial(n,m), m=1..2)-n^2,n=2..51); # Zerinvary Lajos, Jun 19 2008
A089594 := n -> (-1)^n*n*(n+1)/2; # Peter Luschny, Jul 08 2011

nn = Range[50]; Accumulate[(-1)^nn*nn^2] (* Jayanta Basu, Jun 06 2013 *)

for(i=1,50, print1(","sum(j=1,i,(-1)^j*j^2)))

a(n)=(-1)^n*n*(n+1)/2 \\ Charles R Greathouse IV, Jul 08 2011

[(-1)^n*binomial(n+1,2) for n in (1..50)] # G. C. Greubel, Mar 31 2021

From R. J. Mathar, Nov 05 2011: (Start)
a(n) = Sum_{i=1..n} (-1)^i*i^2 = (-1)^n*n*(n+1)/2.
G.f.: -x / (1+x)^3. (End)
a(n) = (-1)^n*det(binomial(i+2,j+1), 1 <= i,j <= n-1). - Mircea Merca, Apr 06 2013
G.f.: -W(0)/(2+2*x), where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) - (k+1)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
E.g.f.: (1/2)*x*(x-2)*exp(-x). - G. C. Greubel, Mar 31 2021
Sum_{n>=1} 1/a(n) = 2 - 4*log(2). - Amiram Eldar, Jan 31 2023

A232600 a(n) = Sum_{k=0..n} k^p*q^k, where p=1, q=-2.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 27 2013

			a(3) = 0^1*2^0 - 1^1*2^1 + 2^1*2^2 - 3^1*2^3 = -18.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-3,0,4).

Cf. A045883, A140960 (absolute values), A059841 (p=0, q=-1), A130472 (p=1 ,q=-1), A089594 (p=2, q=-1), A232599 (p=3, q=-1), A126646 (p=0, q=2), A036799 (p=1, q=2), A036800 (p=q=2), A036827 (p=3, q=2), A077925 (p=0, q=-2), A232601 (p=2, q=-2), A232602 (p=3, q=-2), A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2).

Cf. A045883.

[2*((-2)^n*(3*n+1) -1)/9: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232600:= n-> 2*((-2)^n*(3*n+1) -1)/9; seq(A232600(n), n=0..30); # G. C. Greubel, Mar 31 2021

Table[2((3n+1)(-2)^n -1)/9, {n, 0, 30}] (* Bruno Berselli, Nov 28 2013 *)

PARI
```
a(n)=-((3*n+1)*(-2)^(n+1)+2)/9;
```

[2*((-2)^n*(3*n+1) -1)/9 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = 2*( (3*n+1)*(-2)^n - 1 )/9.
abs(a(n)) = 2*A045883(n) = A140960(n).
From Bruno Berselli, Nov 28 2013: (Start)
G.f.: -2*x / ((1 - x)*(1 + 2*x)^2). [corrected by Georg Fischer, May 11 2019]
a(n) = -3*a(n-1) +4*a(n-3). (End)
From G. C. Greubel, Mar 31 2021: (Start)
E.g.f.: (2/9)*(-exp(x) + (1-6*x)*exp(-2*x)).
a(n) = 2*(-1)^n*A045883(n). (End)

A232601 a(n) = Sum_{k=0..n} k^p*q^k for p = 2 and q = -2.

Original entry on oeis.org

0, -2, 14, -58, 198, -602, 1702, -4570, 11814, -29658, 72742, -175066, 414758, -969690, 2241574, -5131226, 11645990, -26233818, 58700838, -130567130, 288863270, -635980762, 1394062374, -3043511258, 6620165158

Offset: 0

Stanislav Sykora, Nov 27 2013

			a(3) = 0^2*2^0 - 1^2*2^1 + 2^2*2^2 - 3^2*2^3 = -58.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Stanislav Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-5,-6,4,8).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

[2*(1 - (-2)^n*(1-6*n-9*n^2))/27: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232601:= n-> 2*(1 - (-2)^n*(1-6*n-9*n^2))/27; seq(A232601(n), n=0..30); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-5,-6,4,8},{0,-2,14,-58},30] (* Harvey P. Dale, Aug 20 2015 *)

S2M2(n)=((-1)^n*2^(n+1)*(9*n^2+6*n-1)+2)/27;
v = vector(10001); for(k=1, #v, v[k]=S2M2(k-1))

[2*(1 - (-2)^n*(1-6*n-9*n^2))/27 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = 2*((-2)^n * (9*n^2 + 6*n - 1) + 1)/27.
G.f.: 2*x*(-1 + 2*x) / ((1-x)*(1+2*x)^3). - R. J. Mathar, Nov 23 2014
E.g.f.: (2/27)*(exp(x) - (1 +30*x -36*x^2)*exp(-2*x)). - G. C. Greubel, Mar 31 2021
a(n) = - 5*a(n-1) - 6*a(n-2) + 4*a(n-3) + 8*a(n-4). - Wesley Ivan Hurt, Mar 31 2021

A232602 a(n) = Sum_{k=0..n} k^p*q^k, where p=3, q=-2.

Original entry on oeis.org

0, -2, 30, -186, 838, -3162, 10662, -33242, 97830, -275418, 748582, -1977306, 5100582, -12897242, 32060454, -78531546, 189903910, -454052826, 1074770982, -2521320410, 5867287590, -13554437082

Offset: 0

Stanislav Sykora, Nov 27 2013

			a(3) = 0^3*2^0 - 1^3*2^1 + 2^3*2^2 - 3^3*2^3 = -186.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-7,-16,-8,16,16).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

[2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27: n in [0..35]]; // G. C. Greubel, Mar 31 2021

A232602:= n-> 2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27; seq(A232602(n), n=0..35); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-7,-16,-8,16,16}, {0,-2,30,-186,838}, 40] (* G. C. Greubel, Mar 31 2021 *)

a(n)=((-1)^n*2^(n+1)*(27*n^3+27*n^2-9*n-3)+6)/81;

[2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27 for n in (0..35)] # G. C. Greubel, Mar 31 2021

a(n) = 2*(1 - (-2)^n*(1 +3*n -9*n^2 -9*n^3))/27.
G.f.: -2*x*(1-8*x+4*x^2) / ( (1-x)*(1+2*x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (2/27)*(exp(x) - (1 +30*x -144*x^2 +72*x^3)*exp(-2*x)). - G. C. Greubel, Mar 31 2021
a(n) = - 7*a(n-1) - 16*a(n-2) - 8*a(n-3) + 16*a(n-4) + 16*a(n-5). - Wesley Ivan Hurt, Mar 31 2021

A343808 Partial sums of A062074.

Original entry on oeis.org

0, 3, 75, 804, 5988, 36363, 193827, 943968, 4303200, 18652107, 77701107, 313483764, 1231813812, 4734541443, 17859008379, 66286569504, 242605938720, 877071559539, 3136507851387, 11108459253540, 39002734461540, 135876065474523, 470021588191155, 1615461644979264

Offset: 0

Sebastian Krüger, Apr 30 2021

			a(3) = 1^3*3^1+2^3*3^2+3^3*3^3 = 3+8*9+27*27 = 804.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-66,162,-189,81).

Cf. A036827, A062074.

CoefficientList[Series[3x(1 +12x +9x^2)/((1-3x)^4*(1-x)), {x, 0, 23}], x] (* Michael De Vlieger, May 01 2021 *)

a(n) = sum(k=0, n, 3^k*k^3); \\ Michel Marcus, Apr 30 2021

a(n) = ((((n<<2 - 6)*n + 12)*n - 11)*3^(n+1) + 33) >> 3; \\ Kevin Ryde, May 01 2021

[((4*n^3-6*n^2+12*n-11)*3^(n+1) +33)/8 for n in (0..30)] # G. C. Greubel, May 18 2022

a(n) = Sum_{k=0..n} 3^k*k^3.
a(n) = Sum_{k=0..n} A062074(k).
G.f.: 3*x*(1 + 12*x + 9*x^2)/((1 - 3*x)^4*(1 - x)). - Stefano Spezia, May 01 2021
a(n) = ((4*n^3-6*n^2+12*n-11)*3^(n+1) + 33) / 8. - Kevin Ryde, May 01 2021
E.g.f.: (3/8)*(11*exp(x) + (-11 + 30*x + 54*x^2 + 108*x^3)*exp(3*x)). - G. C. Greubel, May 18 2022
a(n) = 13*a(n-1) - 66*a(n-2) + 162*a(n-3) - 189*a(n-4) + 81*a(n-5). - Wesley Ivan Hurt, Apr 16 2023

A368479 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} 2^j * j^k.

Original entry on oeis.org

1, 0, 3, 0, 2, 7, 0, 2, 10, 15, 0, 2, 18, 34, 31, 0, 2, 34, 90, 98, 63, 0, 2, 66, 250, 346, 258, 127, 0, 2, 130, 714, 1274, 1146, 642, 255, 0, 2, 258, 2074, 4810, 5274, 3450, 1538, 511, 0, 2, 514, 6090, 18458, 24810, 19098, 9722, 3586, 1023

Offset: 0

Seiichi Manyama, Dec 26 2023

			Square array begins:
    1,   0,    0,     0,      0,      0,       0, ...
    3,   2,    2,     2,      2,      2,       2, ...
    7,  10,   18,    34,     66,    130,     258, ...
   15,  34,   90,   250,    714,   2074,    6090, ...
   31,  98,  346,  1274,   4810,  18458,   71626, ...
   63, 258, 1146,  5274,  24810, 118458,  571626, ...
  127, 642, 3450, 19098, 107754, 616122, 3557610, ...

OEIS Wiki, Eulerian polynomials.

Columns k=0..3 give A126646, A036799, A036800, A036827.
Main diagonal gives A368466.
Cf. A103438, A173018, A368486.

PARI
```
T(n, k) = sum(j=0, n, 2^j*j^k);
```

G.f. of column k: 2*x*A_k(2*x)/((1-x) * (1-2*x)^(k+1)), where A_n(x) are the Eulerian polynomials for k > 0.

cp's OEIS Frontend

A036828 A036827/2.

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A130472 A permutation of the integers: a(n) = (-1)^n * floor( (n+1)/2 ).

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A232599 Alternating sum of cubes, i.e., Sum_{k=0..n} k^p*q^k for p=3, q=-1.

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A089594 Alternating sum of squares to n.

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A232600 a(n) = Sum_{k=0..n} k^p*q^k, where p=1, q=-2.

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A232601 a(n) = Sum_{k=0..n} k^p*q^k for p = 2 and q = -2.

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