A001788 -id:A001788 - cp's OEIS frontend

A048482 a(n) = T(n,n), array T given by A048472.

1, 3, 13, 49, 161, 481, 1345, 3585, 9217, 23041, 56321, 135169, 319489, 745473, 1720321, 3932161, 8912897, 20054017, 44826625, 99614721, 220200961, 484442113, 1061158913, 2315255809, 5033164801, 10905190401, 23555211265, 50734301185, 108984795137

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A002999, A003013, A001787, A005183, A001788.

[(n^2+n)*2^(n-1) + 1: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

LinearRecurrence[{7,-18,20,-8},{1,3,13,49},30] (* Harvey P. Dale, Feb 02 2015 *)

Vec(-(8*x^3-10*x^2+4*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Nov 26 2014

a(n) = 2 * A001788(n) + 1.
a(n) = (n^2+n)*2^(n-1) + 1. - Ralf Stephan, Sep 02 2003
G.f.: -(8*x^3-10*x^2+4*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Nov 26 2014
a(0)=1, a(1)=3, a(2)=13, a(3)=49, a(n)=7*a(n-1)-18*a(n-2)+ 20*a(n-3)- 8*a(n-4). - Harvey P. Dale, Feb 02 2015

A048506 a(n) = T(0,n), array T given by A048505.

Original entry on oeis.org

1, 2, 7, 25, 81, 241, 673, 1793, 4609, 11521, 28161, 67585, 159745, 372737, 860161, 1966081, 4456449, 10027009, 22413313, 49807361, 110100481, 242221057, 530579457, 1157627905, 2516582401, 5452595201, 11777605633, 25367150593, 54492397569, 116769423361

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is (1, 4, 9, 16, 25, ...).

Similar to A000697 in so far as it can be seen as the transform of 1, 1, 4, 9, 16, ... by a variant of the boustrophedon algorithm (see the Sage implementation). - Peter Luschny, Oct 30 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A048505, A000697.

[n*(n+1)*2^(n-2) + 1: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011

LinearRecurrence[{7,-18,20,-8}, {1,2,7,25}, 30] (* Jean-François Alcover, Jun 11 2019 *)

Vec(-(8*x^3-11*x^2+5*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Nov 26 2014

def sq():
    yield 1
    for n in PositiveIntegers():
        yield n*n
def bous_variant(f):
    k = 0
    am = next(f)
    a = [am]
    while True:
        yield am
        am = next(f)
        a.append(am)
        for j in range(k,-1,-1):
            am += a[j]
            a[j] = am
        k += 1
b = bous_variant(sq())
print([next(b) for  in range(26)]) # _Peter Luschny, Oct 30 2014

a(n) = n*(n+1)*2^(n-2) + 1 = A001788(n) + 1. - Ralf Stephan, Jan 16 2004
a(n) = 7*a(n-1)-18*a(n-2)+20*a(n-3)-8*a(n-4). - Colin Barker, Nov 26 2014
G.f.: -(8*x^3-11*x^2+5*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Nov 26 2014

A052481 a(n) = 2^n*(binomial(n,2) + 1).

Original entry on oeis.org

1, 2, 8, 32, 112, 352, 1024, 2816, 7424, 18944, 47104, 114688, 274432, 647168, 1507328, 3473408, 7929856, 17956864, 40370176, 90177536, 200278016, 442499072, 973078528, 2130706432, 4647288832, 10099884032, 21877489664, 47244640256, 101737037824, 218506461184

Offset: 0

N. J. A. Sloane, Mar 16 2000

a(n) is the generalized Euler number of an (n+2)-dimensional hypercube: (number of vertices) - (number of edges) + (number of faces) = A000079(n+2) - A001787(n+2) + A001788(n+1). - Amiram Eldar, Nov 08 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jonathan F. Mason and Richard H. Hudson, A Generalization of Euler's Formula and its Connection to Fibonacci Numbers, in: Frederic T. Howard (ed.), Applications of Fibonacci Numbers, Volume 9: Proceedings of The Tenth International Research Conference on Fibonacci Numbers and Their Applications, Springer, Dordrecht, 2004, pp. 177-185, alternative link.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A000079, A001787, A001788.

List([0..30], n-> 2^(n-1)*(n^2-n+2)); # G. C. Greubel, May 16 2019

[2^n*(Binomial(n,2)+1): n in [0..30]]; // Vincenzo Librandi, Dec 22 2016

Table[2^n (Binomial[n, 2]+1), {n,0,30}] (* Vincenzo Librandi, Dec 22 2016 *)
LinearRecurrence[{6,-12,8},{1,2,8},30] (* Harvey P. Dale, May 16 2019 *)

{a(n) = 2^(n-1)*(n^2-n+2)}; \\ G. C. Greubel, May 16 2019

[2^(n-1)*(n^2-n+2) for n in (0..30)] # G. C. Greubel, May 16 2019

For the sequence 1, 1, 1, 2, 8, 32, ... we have a(n) = 2^n*(n^2-5n+8)/8. - Paul Barry, Jun 26 2003
From R. J. Mathar, Jan 04 2011: (Start)
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3).
G.f.: (1-4*x+8*x^2)/(1-2*x)^3. (End)
E.g.f.: (1 + 2*x^2)*exp(2*x). - G. C. Greubel, May 16 2019

A056468 a(n) = Sum_{k=1..n} k^6*binomial(n,k).

Original entry on oeis.org

0, 1, 66, 924, 7400, 44040, 217392, 942592, 3714048, 13593600, 46914560, 154328064, 487778304, 1490384896, 4423372800, 12801146880, 36235378688, 100580917248, 274361352192, 736775372800, 1950815354880, 5099601002496, 13176144920576, 33682341494784

Offset: 0

Benoit Cloitre, Dec 06 2002

Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Cf. A001788, A059338, A058649, A058645.

Table[Sum[k^6*Binomial[n, k], {k, n}], {n, 0, 30}] (* T. D. Noe, Nov 22 2013 *)

a(n) = sum(k = 1, n, k^6*binomial(n,k)); \\ Michel Marcus, Nov 20 2013

a(n) = 2^(n-6)*n*(n+1)*(n^4 + 14*n^3 + 31*n^2 - 46*n + 16).

G.f.: -x*(136*x^4-272*x^3+84*x^2+52*x+1)/(2*x-1)^7. [Colin Barker, Sep 20 2012]

A082150 A transform of C(n,2).

Original entry on oeis.org

0, 0, 1, 9, 60, 360, 2040, 11088, 58240, 297216, 1480320, 7223040, 34636800, 163657728, 763549696, 3523645440, 16107110400, 73016672256, 328570011648, 1468890021888, 6528375193600, 28862235279360, 126993714118656

Offset: 0

Paul Barry, Apr 07 2003

Represents the mean of the first and third binomial transforms of C(n,2) Binomial transform of A082149.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-132,504,-1056,1152,-512).

Cf. A082151, A038845, A001788, A000217.

List([0..23], n-> Binomial(n,2)*(2^(n-2)+4^(n-2))/2); # Muniru A Asiru, Feb 12 2018

[Binomial(n,2)*(2^(n-2) + 4^(n-2))/2: n in [0..30]]; // G. C. Greubel, Feb 10 2018

A082150:=[seq(binomial(n,2)*(2^(n-2)+4^(n-2))/2,n=0..23)]; # Muniru A Asiru, Feb 12 2018

CoefficientList[Series[(x^2/(1-2*x)^3 + x^2/(1-4*x)^3)/2, {x,0,50}], x] (* or *) Table[Binomial[n,2]*(2^(n-2) + 4^(n-2))/2, {n,0,30}] (* G. C. Greubel, Feb 10 2018 *)
LinearRecurrence[{18,-132,504,-1056,1152,-512},{0,0,1,9,60,360},30] (* Harvey P. Dale, Jan 17 2022 *)

makelist(2^(n-4)*(2^(n-2)+1)*(n-1)*n, n, 0, 30); /* Bruno Berselli, Feb 13 2018 */

for(n=0,30, print1(binomial(n,2)*(2^(n-2) + 4^(n-2))/2, ", ")) \\ G. C. Greubel, Feb 10 2018

a(n) = C(n, 2)*(2^(n-2) + 4^(n-2))/2.
G.f.: (x^2/(1-2*x)^3 + x^2/(1-4*x)^3)/2.
G.f.: x^2*(36*x^3 - 30*x^2 + 9*x-1)/((1 - 2*x)^3*(4*x - 1)^3).
E.g.f.: x^2*exp(3*x)*cosh(x)/2.
From Bruno Berselli, Feb 12 2018: (Start)
E.g.f.: x^2*(1 + exp(2*x))*exp(2*x)/4.
a(n) = 2^(n-4)*(2^(n-2) + 1)*(n - 1)*n. (End)

A087076 Sums of the squares of the elements in the subsets of the integers 1 to n.

Original entry on oeis.org

0, 1, 10, 56, 240, 880, 2912, 8960, 26112, 72960, 197120, 518144, 1331200, 3354624, 8314880, 20316160, 49020928, 116981760, 276430848, 647495680, 1504706560, 3471835136, 7958691840, 18136170496, 41104179200, 92694118400, 208071032832

Offset: 0

Alan Sutcliffe (alansut(AT)ntlworld.com), Aug 08 2003

A000076 gives the number of subsets of 1 to n. A001787 gives the number of elements in the subsets. A001788 gives the sum of the elements.

			a(3)=56 since the subsets of (1,2,3) are ( ) (1) (1,2) (1,3) (1,2,3) (2) (2,3) (3) and the sum of the squares of the elements in these subsets is 56.

Matthew House, Table of n, a(n) for n = 0..3272
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A058645 has the same then similar initial values.

Equals A014483 shifted right.

a(n) = (2^(n-2))*n*(n+1)*(2*n+1)/3 \\ Michel Marcus, Jul 12 2013

a(n) = 2^(n - 2)*n*(n + 1)*(2*n + 1)/3.
G.f.: x*(1 + 2*x)/(1 - 2*x)^4.
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4). - Matthew House, Feb 13 2017
a(n) = (1/2) * Sum_{k=0..n} Sum_{i=0..n} i^2 * C(n,k). - Wesley Ivan Hurt, Sep 21 2017

A130811 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 5-subsets of X containing none of X_i, (i=1,...n).

Original entry on oeis.org

32, 192, 672, 1792, 4032, 8064, 14784, 25344, 41184, 64064, 96096, 139776, 198016, 274176, 372096, 496128, 651168, 842688, 1076768, 1360128, 1700160, 2104960, 2583360, 3144960, 3800160, 4560192, 5437152, 6444032, 7594752, 8904192

Offset: 5

Milan Janjic, Jul 16 2007

Number of n permutations (n>=5) of 3 objects u,v,z, with repetition allowed, containing n-5 u's. Example: if n=5 then n-5 =(0) zero u, a(1)=32. - Zerinvary Lajos, Aug 05 2008

a(n) is the number of 4-dimensional elements in an n-cross polytope where n>=5. - Patrick J. McNab, Jul 06 2015

Milan Janjic, Two Enumerative Functions
Eric Weisstein's World of Mathematics, Cross Polytope

Cf. A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809, A130810. - Zerinvary Lajos, Aug 05 2008

[Binomial(n,n-5)*2^5: n in [5..40]]; // Vincenzo Librandi, Jul 09 2015

a:=n->binomial(2*n,5)+(2*n-4)*binomial(n,2)-n*binomial(2*n-2,3)
seq(binomial(n,n-5)*2^5,n=5..34); # Zerinvary Lajos, Dec 07 2007
seq(binomial(n+4, 5)*2^5, n=1..22); # Zerinvary Lajos, Aug 05 2008

Table[Binomial[2 n, 5] + (2 n - 4) Binomial[n, 2] - n Binomial[2 n - 2, 3], {n, 5, 40}] (* Vincenzo Librandi, Jul 09 2015 *)

a(n) = binomial(2*n,5) + (2*n-4)*binomial(n,2) - n*binomial(2*n-2,3).
a(n) = C(n,n-5)*2^5, for n>=5. - Zerinvary Lajos, Dec 07 2007
G.f.: 32*x^5/(1-x)^6. - Colin Barker, Apr 14 2012

A181407 a(n) = (n-4)(n-3)2^(n-2).

Original entry on oeis.org

3, 3, 2, 0, 0, 16, 96, 384, 1280, 3840, 10752, 28672, 73728, 184320, 450560, 1081344, 2555904, 5963776, 13762560, 31457280, 71303168, 160432128, 358612992, 796917760, 1761607680, 3875536896, 8489271296, 18522046464, 40265318400, 87241523200, 188441690112

Offset: 0

Paul Curtz, Jan 28 2011

Binomial transform of (3, 0, -1, followed by A005563).
The sequence and its successive differences are:
   3,  3,  2,   0,   0,   16,   96,  384,      a(n),
   0, -1, -2,   0,  16,   80,  288,  896,      A178987,
  -1, -1,  2,  16,  64,  208,  608, 2688,     -A127276,
   0,  3, 14,  48, 144,  400, 1056, 2688,      A176027,
   3, 11, 34,  96, 256,  656, 1632, 3968,      A084266(n+1)
   8, 23, 62, 160, 400,  976, 2336, 5504,
  15, 39, 98, 240, 576, 1360, 3168, 7296.
Division of the k-th column by abs(A174882(k)) gives
   3,  3,  1,  0,  0,  1,  3,  3,  5,  15,  21, 14,   A064038(n-3),
   0, -1, -1,  0,  1,  5,  9,  7, 10,  27,  35, 22,   A160050(n-3),
  -1, -1,  1,  2,  4, 13, 19, 13, 17,  43,  53, 32,   A176126,
   0,  3,  7,  6,  9, 25, 33, 21, 26,  63,  75, 44,   A178242,
   3, 11, 17, 12, 16, 41, 51, 31, 37,  87, 101, 58,
   8  23, 31, 20, 25, 61, 73, 43, 50, 115, 131, 74,
  15, 39, 49, 30, 36, 85, 99, 57, 65, 147, 165, 92.
Columns are (or from) A005563, A142463, A056220, A002378, A000290, A001844, A058331, A002061, A002522, A097080, A093328, A014206.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A176027, A181318.

List([0..40], n-> (n-4)*(n-3)*2^(n-2)); # G. C. Greubel, Feb 21 2019

[(n-4)*(n-3)*2^(n-2): n in [0..40] ]; // Vincenzo Librandi, Feb 01 2011

Table[(n-4)*(n-3)*2^(n-2), {n,0,40}] (* G. C. Greubel, Feb 21 2019 *)

vector(40, n, n--; (n-4)*(n-3)*2^(n-2)) \\ G. C. Greubel, Feb 21 2019

[(n-4)*(n-3)*2^(n-2) for n in (0..40)] # G. C. Greubel, Feb 21 2019

a(n) = 16*A001788(n-4).
a(n+1) - a(n) = A178987(n).
G.f.: (3 - 15*x + 20*x^2) / (1-2*x)^3. - R. J. Mathar, Jan 30 2011
E.g.f.: (x^2 - 3*x + 3)*exp(2*x). - G. C. Greubel, Feb 21 2019

A190050 Expansion of ((1-x)(3x^2-3x+1))/(1-2x)^3.

Original entry on oeis.org

1, 2, 6, 17, 46, 120, 304, 752, 1824, 4352, 10240, 23808, 54784, 124928, 282624, 634880, 1417216, 3145728, 6946816, 15269888, 33423360, 72876032, 158334976, 342884352, 740294656, 1593835520, 3422552064

Offset: 0

Johannes W. Meijer, May 06 2011

The second left hand column of triangle A175136.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A175136, A011782, A190951.

Related to A001788.

[1] cat [(n^2 + 5*n + 10)*2^(n-4): n in [1..30]]; // G. C. Greubel, Jan 10 2018

A190050:= proc(n) option remember; if n=0 then A190050(n):=1: else A190050(n):=(n^2+5*n+10)*2^(n-4) fi: end: seq (A190050(n), n=0..26);

Join[{1}, LinearRecurrence[{6,-12,8}, {2,6,17}, 30]] (* or *) CoefficientList[Series[((1-x)*(3*x^2-3*x+1))/(1-2*x)^3, {x, 0, 50}], x] (* G. C. Greubel, Jan 10 2018 *)

x='x+O('x^30); Vec(((1-x)*(3*x^2-3*x+1))/(1-2*x)^3) \\ G. C. Greubel, Jan 10 2018

for(n=0,30, print1(if(n==0,1,(n^2 + 5*n + 10)*2^(n-4)), ", ")) \\ G. C. Greubel, Jan 10 2018

G.f.: ((1-x)*(3*x^2-3*x+1))/(1-2*x)^3.
a(n) = (n^2 + 5*n + 10)*2^(n-4) for n >=1 with a(0)=1.
a(n) = A001788(n+1) -4*A001788(n) +6*A001788(n-1) -3*A001788(n-2) for n >=1 with a(0)=1.

A213344 2-quantum transitions in systems of N>=2 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 6, 24, 4, 80, 40, 240, 240, 15, 672, 1120, 210, 1792, 4480, 1680, 56, 4608, 16128, 10080, 1008, 11520, 53760, 50400, 10080, 210, 28160, 168960, 221760, 73920, 4620, 67584, 506880, 887040, 443520, 55440, 792

Offset: 2

Stanislav Sykora, Jun 09 2012

For a general discussion, please see A213343.
This a(n) is for double-quantum transitions (q = 2).
It lists the flattened triangle T(2;N,k) with rows N = 2,3,... and columns N, k = 0..floor((N-2)/2).

			For N=4, there are 4 second-quantum transitions with combination index 1: (0001,1110),(0010,1101),(0100,1011),(1000,0111).
Starting rows of the triangle:
  N | k = 0, 1, ..., floor((N-2)/2)
  2 |   1
  3 |   6
  4 |  24   4
  5 |  80  40
  6 | 240 240 15

See A213343.

Stanislav Sykora, Table of n, a(n) for n = 2..2501
Stanislav Sykora, T(2;N,k) with rows N=2,..,100 and columns k=0,..,floor((N-2)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 (q=1), A213345 to A213352 (q=3..10).

Cf. A001788 (first column), A002694 (row sums).

With[{q = 2}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 12}, {k, 0, Floor[(n - 2)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

PARI
```
See A213343; set thisq = 2.
```

Set q = 2 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

cp's OEIS Frontend

A048482 a(n) = T(n,n), array T given by A048472.

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A048506 a(n) = T(0,n), array T given by A048505.

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A052481 a(n) = 2^n*(binomial(n,2) + 1).

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A056468 a(n) = Sum_{k=1..n} k^6*binomial(n,k).

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A082150 A transform of C(n,2).

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A087076 Sums of the squares of the elements in the subsets of the integers 1 to n.

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A130811 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 5-subsets of X containing none of X_i, (i=1,...n).

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A181407 a(n) = (n-4)(n-3)2^(n-2).

A190050 Expansion of ((1-x)(3x^2-3x+1))/(1-2x)^3.