A069996 -id:A069996 - cp's OEIS frontend

A153703 Partial sums of A069996.

1, 13, 94, 526, 2551, 11299, 47020, 186988, 718429, 2686729, 9831658, 35340826, 125154355, 437641663, 1513809688, 5187129880, 17627632249, 59469045061, 199327841590, 664232428390, 2201904349231, 7264715299483, 23865295832644, 78091766836996

Offset: 1

Bruno Berselli, Dec 12 2010

The first differences are in the third row of the square array of A072590.

The general formula for the partial sums of the sequence 1, 4*m, 9*m^2, 16*m^3, 25*m^4,...,n^2*m^(n-1),... is (n^2*m^(n+2)-(2*n*(n+1)-1)*m^(n+1)+(n+1)^2*m^n-m-1)/(m-1)^3 with m>1 (see also References).

"Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno) - Apr / May, 1913 - p. 99 (Problem 1277, case x=3).

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (10,-36,54,-27).

Cf. A069996, A072590, A027472, A003462.

[(3^n*(n^2-n+1)-1)/2: n in [1..25]]; // Vincenzo Librandi, Aug 19 2013

CoefficientList[Series[(1 + 3 x) / ((1 - x) (1 - 3 x)^3), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 19 2013 *)

a(n) = (3^n*(n^2-n+1)-1)/2 \\ Michel Marcus, Jun 07 2013

a(n) = (3^n*(n^2 - n + 1) - 1)/2.
G.f.: x*(1+3*x)/((1-x)*(1-3*x)^3).
a(n) = 10*a(n-1) - 36*a(n-2) + 54*a(n-3) - 27a(n-4) for n>4.
a(n) = 9*A027472(n+1) + A003462(n) for n>2.
E.g.f.: (1/2)*((1 + 9*x^2)*exp(x) - exp(-x))*exp(2*x). - G. C. Greubel, Aug 24 2016

A072590 Table T(n,k) giving number of spanning trees in complete bipartite graph K(n,k), read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 32, 81, 32, 1, 1, 80, 432, 432, 80, 1, 1, 192, 2025, 4096, 2025, 192, 1, 1, 448, 8748, 32000, 32000, 8748, 448, 1, 1, 1024, 35721, 221184, 390625, 221184, 35721, 1024, 1, 1, 2304, 139968, 1404928, 4050000, 4050000

Offset: 1

Michael Somos, Jun 23 2002

			From _Andrew Howroyd_, Oct 29 2019: (Start)
Array begins:
============================================================
n\k | 1   2     3       4        5         6           7
----+-------------------------------------------------------
  1 | 1   1     1       1        1         1           1 ...
  2 | 1   4    12      32       80       192         448 ...
  3 | 1  12    81     432     2025      8748       35721 ...
  4 | 1  32   432    4096    32000    221184     1404928 ...
  5 | 1  80  2025   32000   390625   4050000    37515625 ...
  6 | 1 192  8748  221184  4050000  60466176   784147392 ...
  7 | 1 448 35721 1404928 37515625 784147392 13841287201 ...
  ...
(End)

J. W. Moon, Counting Labeled Trees, Issue 1 of Canadian mathematical monographs, Canadian Mathematical Congress, 1970.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 5.66.

T. D. Noe, Antidiagonals d=1..50, flattened
Taylor Brysiewicz and Aida Maraj, Lawrence Lifts, Matroids, and Maximum Likelihood Degrees, arXiv:2310.13064 [math.CO], 2023. See p. 13.
H. I. Scoins, The number of trees with nodes of alternate parity, Proc. Cambridge Philos. Soc. 58 (1962) 12-16.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Spanning Tree

Columns 2..3 are A001787, A069996.
Main diagonal is A068087.
Antidiagonal sums are A132609.
Cf. A070285, A328887, A328888.

t[n_, k_] := n^(k-1) * k^(n-1); Table[ t[n-k+1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 21 2013 *)

{T(n, k) = if( n<1 || k<1, 0, n^(k-1) * k^(n-1))}

T(n, k) = n^(k-1) * k^(n-1).

E.g.f.: A(x,y) - 1, where: A(x,y) = exp( x*exp( y*A(x,y) ) ) = Sum_{n>=0} Sum_{k=0..n} (n-k)^k * (k+1)^(n-k-1) * x^(n-k)/(n-k)! * y^k/k!. - Paul D. Hanna, Jan 22 2019

Scoins reference from Philippe Deléham, Dec 22 2003

A068087 a(n) = n^(2*n-2).

Original entry on oeis.org

1, 4, 81, 4096, 390625, 60466176, 13841287201, 4398046511104, 1853020188851841, 1000000000000000000, 672749994932560009201, 552061438912436417593344, 542800770374370512771595361, 629983141281877223603213172736, 852226929923929274082183837890625

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 06 2002

Number of spanning trees in the bipartite graph K(n,n). In general the number of spanning trees in the bipartite graph K(m,n) is m^(n-1) * n^(m-1).

Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Spanning Tree

a(n) = A000169(n)^2.

Cf. A069996, A001787, A072590, A294360.

a(n)=n^(2*n-2) \\ Charles R Greathouse IV, Mar 31 2016

A084857 Inverse binomial transform of n^2*3^(n-1).

Original entry on oeis.org

0, 1, 10, 48, 176, 560, 1632, 4480, 11776, 29952, 74240, 180224, 430080, 1011712, 2351104, 5406720, 12320768, 27852800, 62521344, 139460608, 309329920, 682622976, 1499463680, 3279945728, 7147094016, 15518924800, 33587986432, 72477573120, 155960999936

Offset: 0

Paul Barry, Jun 09 2003

Binomial transform of octagonal numbers A000567. Inverse binomial transform of A069996.

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A000567, A069996.

LinearRecurrence[{6,-12,8},{0,1,10},30] (* Harvey P. Dale, Dec 28 2019 *)

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

O.g.f.: -x*(1+4*x)/(-1+2*x)^3. - R. J. Mathar, Apr 02 2008

A127960 a(n) = n^2*3^n.

Original entry on oeis.org

0, 3, 36, 243, 1296, 6075, 26244, 107163, 419904, 1594323, 5904900, 21434787, 76527504, 269440587, 937461924, 3228504075, 11019960576, 37321507107, 125524238436, 419576389587, 1394713760400, 4613015762523, 15188432850756

Offset: 0

Mohammad K. Azarian, Apr 07 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A036289, A007758, A069996.

[n^2*3^n: n in [0..30]]; // Vincenzo Librandi, Feb 07 2013

LinearRecurrence[{9,-27,27}, {0,3,36}, 25] (* or *) CoefficientList[ Series[3*x(1 + 3*x)/(1 - 3*x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 07 2013 *)

for(n=0,30, print1(n^2*3^n, ", ")) \\ G. C. Greubel, May 04 2018

G.f.: 3*x*(1 + 3*x)/(1 - 3*x)^3. - Vincenzo Librandi, Feb 07 2013
a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3). - Vincenzo Librandi, Feb 07 2013
a(n) = 3*A069996(n) for n>0. - Bruno Berselli, Feb 07 2013
E.g.f.: (9*x^2 + 3*x)*exp(3*x). - G. C. Greubel, May 04 2018

A278417 a(n) = n*((2+sqrt(3))^n + (2-sqrt(3))^n)/2.

Original entry on oeis.org

0, 2, 14, 78, 388, 1810, 8106, 35294, 150536, 632034, 2620870, 10759342, 43804812, 177105266, 711809378, 2846259390, 11330543632, 44929049794, 177540878718, 699402223118, 2747583822740, 10766828545746, 42095796462874, 164244726238366, 639620518118424, 2486558615814050, 9651161613824822, 37403957244654702

Offset: 0

Indranil Ghosh, Nov 21 2016

This was originally based on a graph theory formula in the Wikipedia which turned out to be wrong.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-18,8,-1).

Cf. A030019, A069996, A139400, A193153.

f:=n->expand(n*((2+sqrt(3))^n + (2-sqrt(3))^n)/2); # N. J. A. Sloane, May 13 2017

Table[Simplify[(n/2) (((2 + #)^n + (2 - #)^n)) &@ Sqrt@ 3], {n, 3, 27}] (* or *)
Drop[#, 3] &@ CoefficientList[Series[2 x^3*(39 - 118 x + 55 x^2 - 7 x^3)/(1 - 4 x + x^2)^2, {x, 0, 27}], x] (* Michael De Vlieger, Nov 24 2016 *)
LinearRecurrence[{8,-18,8,-1},{0,2,14,78},30] (* Harvey P. Dale, Jan 01 2021 *)

vector(25, n, n+=2; n*((2+sqrt(3))^n + ((2-sqrt(3))^n))/2) \\ Colin Barker, Nov 21 2016

Vec(2*x^3*(39 - 118*x + 55*x^2 - 7*x^3) / (1 - 4*x + x^2)^2 + O(x^30)) \\ Colin Barker, Nov 21 2016

def a278417(n):
    a = [0, 2, 14, 78, 388, 1810]
    if n < 6:
        return a[n]
    for k in range(n - 5):
        a = a[1:] + [7*a[-1] - 10*a[-2] - 10*a[-3] + 7*a[-4] - a[-5]]
    return a[-1]
# David Radcliffe, May 09 2025

From Colin Barker, Nov 21 2016: (Start)
a(n) = 7*a(n-1) - 10*a(n-2) - 10*a(n-3) + 7*a(n-4) - a(n-5) for n>6.
G.f.: 2*x^3*(39 - 118*x + 55*x^2 - 7*x^3) / (1 - 4*x + x^2)^2.
(End)

Entry revised by N. J. A. Sloane, May 13 2017

A334062 Triangle read by rows: T(n,k) is the number of non-crossing set partitions of {1..4n} into n sets of 4 with k of the sets being a contiguous set of elements.

Original entry on oeis.org

1, 3, 1, 9, 12, 1, 27, 81, 31, 1, 81, 432, 390, 65, 1, 243, 2025, 3330, 1365, 120, 1, 729, 8748, 22815, 17415, 3909, 203, 1, 2187, 35721, 135513, 166320, 70938, 9730, 322, 1, 6561, 139968, 728028, 1312038, 911358, 242004, 21816, 486, 1, 19683, 531441, 3630420, 9032310, 9294264, 4067658, 722316, 45090, 705, 1

Offset: 1

Donovan Young, May 28 2020

T(n,k) is also the number of non-crossing configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 4n, see [Young].
For the case of partitions of {1..3n} into sets of 3, see A091320.
For the case of partitions of {1..2n} into sets of 2, see A001263.

			Triangle starts:
    1;
    3,    1;
    9,   12,    1;
   27,   81,   31,    1;
   81,  432,  390,   65,   1;
  243, 2025, 3330, 1365, 120, 1;
  ...
For n=2 and k=1 the configurations are (1,6,7,8),(2,3,4,5), (1,2,7,8),(3,4,5,6), and (1,2,3,8),(4,5,6,7); hence T(2,1) = 3.

Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020.

Row sums are A002293.
Column 2 is A069996.
Cf. A001263, A091320, A334063.

G.f.: G(t, z) satisfies z*G^4 - (1 + z - t*z)*G + 1 = 0.

cp's OEIS Frontend

A153703 Partial sums of A069996.

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A072590 Table T(n,k) giving number of spanning trees in complete bipartite graph K(n,k), read by antidiagonals.

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A068087 a(n) = n^(2*n-2).

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A084857 Inverse binomial transform of n^2*3^(n-1).

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A127960 a(n) = n^2*3^n.

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A278417 a(n) = n*((2+sqrt(3))^n + (2-sqrt(3))^n)/2.

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A334062 Triangle read by rows: T(n,k) is the number of non-crossing set partitions of {1..4n} into n sets of 4 with k of the sets being a contiguous set of elements.

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