A038845 -id:A038845 - cp's OEIS frontend

A111636 Triangle read by rows: T(n,k) (0<=k<=n) is the number of labeled graphs having k blue nodes and n-k green ones and only nodes of different colors can be joined by an edge.

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 32, 96, 32, 1, 1, 80, 640, 640, 80, 1, 1, 192, 3840, 10240, 3840, 192, 1, 1, 448, 21504, 143360, 143360, 21504, 448, 1, 1, 1024, 114688, 1835008, 4587520, 1835008, 114688, 1024, 1, 1, 2304, 589824, 22020096, 132120576, 132120576, 22020096, 589824, 2304, 1

Offset: 0

Emeric Deutsch, Aug 09 2005

Row sums yield A047863. T(2*n,n) = A111637(n). T(n,1) = A001787(n).

			T(2,1)=4 because we have B G, B--G, G B and G--B, where B (G) stands for a blue (green) node and -- denotes an edge.
Triangle starts:
  1;
  1,  1;
  1,  4,  1;
  1, 12, 12,  1;
  1, 32, 96, 32, 1;
  ...

H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 88, Eq. 3.11.2.

S. R. Finch, Bipartite, k-colorable and k-colored graphs
S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A047863, A111637, A001787.
Cf. A134530 (matrix log), A134531.
Cf. A000684, A011266, A038845, A140802, A224069 (matrix inverse).

T:=(n,k)->binomial(n,k)*2^(k*(n-k)): for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

nn=6;f[x_,y_]:=Sum[Exp[x 2^n](y x)^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[f[x,y],{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Sep 07 2013 *)

T(n, k)=2^[k(n-k)]*C(n, k).
Matrix log yields triangle A134530, where A134530(n,k) = A134531(n-k)*(2^k)^(n-k)*C(n,k). - Paul D. Hanna, Nov 11 2007
From Peter Bala, Aug 13 2012: (Start)
Let f(n) = n!*2^binomial(n,2) = A011266(n). Then T(n,k) = f(n)/(f(k)*f(n-k)).
E.g.f.: Sum_{n >= 0} exp(2^n*t*x)*x^n/n! = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + ....
O.g.f.: Sum_{n >= 0} x^n/(1-2^n*t*x)^(n+1) = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... O.g.f. for column k: 1/(1-2^k*x)^(k+1).
Recurrence equation: T(n,k) = 2^k*T(n-1,k) + 2^(n-k)*T(n-1,k-1).
Column k = 2: A038845. Column k = 3: A140802. Sum_{k = 0..n} k*T(n,k) = n*A000684(n). (End)
From Peter Bala, Apr 09 2013: (Start)
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + .... Then a generating function for this sequence is E(z)*E(x*z) = 1 + (1 + x)*z + (1 + 4*x + x^2)*z^2/(2!*2) + (1 + 12*x + 12*x^2 + x^3)*z^3/(3!*2^3) + .... Cf. Pascal's triangle A007318 with an e.g.f. of exp(z)*exp(x*z).
This is a generalized Riordan array (E(x), x) with respect to the sequence n!*2^C(n,2), as defined by Wang and Wang.
The n-th power of this triangle has a generating function E(z)^n*E(x*z). See A224069 for the inverse array (n = -1).
The n-th row is a log-concave sequence and hence unimodal.
The row polynomials satisfy the recurrence equation R(n+1,x) = 2^n*x*R(n,x/2) + R(n,2*x) with R(0,x) = 1, as well as R'(n,2*x) = n*2^(n-1)*R(n-1,x) (the ' denotes differentiation w.r.t. x). The row polynomials appear to have only real zeros.
Sum_{k = 0..n} (-1)^k*T(2*n+1,k) = 0;
Sum_{k = 0..n} (-1)^k*2^k*T(2*n,k) = 0;
Sum_{k = 0..n} 2^k*T(n,k) = A000684(n). (End)
T(n,k+1) = Product_{i=0..k} (T(n-i,1)/T(i+1,1)) for 0 <= k < n. - Werner Schulte, Nov 13 2018

A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

Original entry on oeis.org

1, 24, 336, 3584, 32256, 258048, 1892352, 12976128, 84344832, 524812288, 3148873728, 18320719872, 103817412608, 574988746752, 3121367482368, 16647293239296, 87398289506304, 452414675091456, 2312341672689664, 11683410556747776, 58417052783738880, 289303499500421120

Offset: 0

Wolfdieter Lang

Also convolution of A020922 with A000984 (central binomial coefficients); also convolution of A040075 with A000302 (powers of 4).
With a different offset, number of n-permutations of 5 objects: u,v,z,x, y with repetition allowed, containing exactly five (5) u's. Example: a(1)=24 because we have uuuuuv uuuuvu uuuvuu uuvuuu uvuuuu vuuuuu uuuuuz uuuuzu uuuzuu uuzuuu uzuuuu zuuuuu uuuuux uuuuxu uuuxuu uuxuuu uxuuuu xuuuuu uuuuuy uuuuyu uuuyuu uuyuuu uyuuuu yuuuuu. - Zerinvary Lajos, Jun 16 2008
Also convolution of A002457 with A020920, also convolution of A002697 with A038846, also convolution of A002802 with A020918, also convolution of A038845 with A038845. - Rui Duarte, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-240,1280,-3840,6144,-4096).

Cf. A038231.

List([0..30], n-> 4^n*Binomial(n+5,5)); # G. C. Greubel, Jul 20 2019

[4^n*Binomial(n+5, 5): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(seq(binomial(i+5, j)*4^i, j =i), i=0..30); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+5,5)*4^n,n=0..30); # Zerinvary Lajos, Jun 16 2008

CoefficientList[Series[1/(1-4x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {24,-240,1280,-3840,6144,-4096}, {1,24,336,3584,32256, 258048}, 30] (* Harvey P. Dale, Mar 24 2018 *)

Vec(1/(1-4*x)^6 + O(x^30)) \\ Michel Marcus, Aug 21 2015

[lucas_number2(n, 4, 0)*binomial(n,5)/2^10 for n in range(5, 35)] # Zerinvary Lajos, Mar 11 2009

a(n) = binomial(n+5, 5)*4^n.
G.f.: 1/(1-4*x)^6.
a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10+i_11+i_12 = n} f(i_1)* f(i_2)*f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10) *f(i_11)*f(i_12), with f(k)=A000984(k). - Rui Duarte, Oct 08 2011
E.g.f.: (15 + 120*x + 240*x^2 + 160*x^3 + 32*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 1620*log(4/3) - 465.
Sum_{n>=0} (-1)^n/a(n) = 12500*log(5/4) - 8365/3. (End)

A045724 Convolution of Catalan numbers A000108 with A020918.

Original entry on oeis.org

1, 15, 142, 1083, 7266, 44758, 259356, 1435347, 7663898, 39761282, 201483204, 1001098462, 4891910100, 23565178380, 112118316088, 527674017411, 2459747256138, 11368724035210, 52145629874100, 237541552456362

Offset: 0

Wolfdieter Lang

Also convolution of A001700 with A038845; also convolution of A029887 with A000302 (powers of 4); also convolution of A042941 with A000984 (central binomial coefficients).

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

Cf. A000108, A000302, A000984, A001700, A020918, A029887, A042941.

[(Binomial(n+5,4)*Catalan(n+4) -5*4^(n+1)*Binomial(n+3,2))/10: n in [0..40]]; // G. C. Greubel, Jul 19 2024

Table[(Binomial[n+5,4]*CatalanNumber[n+4] -5*4^(n+1)*Binomial[n+3,2] )/10, {n,0,40}] (* G. C. Greubel, Jul 19 2024 *)

[(binomial(n+5,4)*catalan_number(n+4) - 5*4^(n+1)*binomial(n+3,2))/10 for n in range(41)] # G. C. Greubel, Jul 19 2024

a(n) = binomial(n+4, 3)*A000984(n+4)/(2*A000984(3)) - (n+3)*(n+2)*4^n, where A000984(n) = binomial(2*n, n),

G.f.: c(x)/(1-4*x)^(7/2) = (2 - c(x))/(1-4*x)^4, where c(x) = g.f. for Catalan numbers.

A367022 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = 4^n * hypergeom([1/2, -n - 1, -n], [2, 2], x).

Original entry on oeis.org

1, 4, 1, 16, 12, 2, 64, 96, 48, 5, 256, 640, 640, 200, 14, 1024, 3840, 6400, 4000, 840, 42, 4096, 21504, 53760, 56000, 23520, 3528, 132, 16384, 114688, 401408, 627200, 439040, 131712, 14784, 429, 65536, 589824, 2752512, 6021120, 6322176, 3161088, 709632, 61776, 1430

Offset: 0

Peter Luschny, Nov 06 2023

			Triangle T(n, k) starts:
  [0]     1;
  [1]     4,      1;
  [2]    16,     12,       2;
  [3]    64,     96,      48,       5;
  [4]   256,    640,     640,     200,      14;
  [5]  1024,   3840,    6400,    4000,     840,      42;
  [6]  4096,  21504,   53760,   56000,   23520,    3528,    132;
  [7] 16384, 114688,  401408,  627200,  439040,  131712,  14784,   429;
  [8] 65536, 589824, 2752512, 6021120, 6322176, 3161088, 709632, 61776, 1430;

Cf. A038845 (column 1), A128088, A005802, A246513, A001263.

p := n -> 4^n*hypergeom([1/2, -n - 1, -n], [2, 2], x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..8);

T[n_,k_]:=4^(n-k)*Binomial[n,k]*Binomial[n+1,k]*Binomial[2*k,k]/(k+1)^2;Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (* Detlef Meya, Nov 20 2023 *)

From Detlef Meya, Nov 20 2023: (Start)
T(n, k) = 4^(n - k)*binomial(n, k)*binomial(n+1, k)*binomial(2*k, k)/(k + 1)^2.
T(n, k) = A001263(n+1, k+1)*4^(n - k)*binomial(2*k, k)/(k + 1). (End)

A129532 3n(n-1)4^(n-2).

Original entry on oeis.org

0, 0, 6, 72, 576, 3840, 23040, 129024, 688128, 3538944, 17694720, 86507520, 415236096, 1962934272, 9160359936, 42278584320, 193273528320, 876173328384, 3942779977728, 17626545782784, 78340203479040, 346346162749440

Offset: 0

Emeric Deutsch, Apr 22 2007

nonn

Number of inversions in all 4-ary words of length n on {0,1,2,3}. Example: a(2)=6 because each of the words 10,20,30,21,31,32 has one inversion and the words 00,01,02,03,11,12,13,22,23,33 have no inversions. a(n)=Sum(k*A129531(n,k),k>=0). a(n)=6*A038845(n-2).

Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A038845, A129531, A001788, A129530.

Maple
```
seq(3*n*(n-1)*4^(n-2),n=0..25);
```

Table[3n(n-1)4^(n-2),{n,0,30}] (* or *) LinearRecurrence[{12,-48,64},{0,0,6},30] (* Harvey P. Dale, May 25 2018 *)

G.f.=6x^2/(1-4x)^3.

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)

A052780 Expansion of e.g.f. x^2exp(4x).

Original entry on oeis.org

0, 0, 2, 24, 192, 1280, 7680, 43008, 229376, 1179648, 5898240, 28835840, 138412032, 654311424, 3053453312, 14092861440, 64424509440, 292057776128, 1314259992576, 5875515260928, 26113401159680

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Old name was: A simple grammar.

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 737
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A038845.

List([0..30], n-> 4^(n-2)*n*(n-1)); # G. C. Greubel, Jul 20 2019

[4^(n-2)*n*(n-1): n in [0..30]]; // G. C. Greubel, Jul 20 2019

spec := [S,{B=Set(Z),S=Prod(Z,Z,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(n*(n-1)*4^(n-2), n=0..20); # Zerinvary Lajos, Apr 25 2007

Table[4^(n-2)*n*(n-1), {n,0,30}] (* G. C. Greubel, Jul 20 2019 *)
With[{nn=20},CoefficientList[Series[x^2 Exp[4x],{x,0,nn}],x] Range[0,nn]!] (* or *) LinearRecurrence[{12,-48,64},{0,0,2},30] (* Harvey P. Dale, Sep 28 2022 *)

vector(30, n, n--; 4^(n-2)*n*(n-1)) \\ G. C. Greubel, Jul 20 2019

[4^(n-2)*n*(n-1) for n in (0..30)] # G. C. Greubel, Jul 20 2019

E.g.f.: x^2*exp(x)^4.
a(n) = 2*A038845(n-2).
Recurrence: a(1)=0, a(2)=2, (n-1)*a(n+1) - 4*(n+1)*a(n) = 0.
From Ralf Stephan, Mar 26 2003: (Start)
a(n) = n*(n-1)*4^(n-2).
G.f.: 2*x^2/(1-4*x)^3. (End)

New name from e.g.f. by Jon E. Schoenfield, Feb 07 2019

A041005 Convolution of Catalan numbers A000108(n+1), n >= 0, with A020918.

Original entry on oeis.org

1, 16, 159, 1260, 8722, 55152, 326811, 1844084, 10015566, 52754624, 270976342, 1362986520, 6734927460, 32775704608, 157408497171, 747269225028, 3511471892470, 16351481223840, 75525932249922, 346305571781224

Offset: 0

Wolfdieter Lang

Also convolution of A001791(n+1), n >= 0, with A038845; also convolution of A008549(n+1), n >= 0, with A002802; also convolution of A029760 with A002697; also convolution of A038806(n+1), n >= 0, with A002457; also convolution of A038836 with A000302 (powers of 4); also convolution of A041001 with A000984 (central binomial coefficients).

a(n)=binomial(n+7, 3)*binomial(2*(n+4), n+2)/20 - (n+4)*(n+3)*4^(n+1); G.f. (c(x)^2)/(1-4*x)^(7/2), where c(x) = g.f. for Catalan numbers.

A116144 a(n) = 4^n * n*(n+1).

Original entry on oeis.org

0, 8, 96, 768, 5120, 30720, 172032, 917504, 4718592, 23592960, 115343360, 553648128, 2617245696, 12213813248, 56371445760, 257698037760, 1168231104512, 5257039970304, 23502061043712, 104453604638720, 461794883665920

Offset: 0

Mohammad K. Azarian, Apr 08 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A007758, A036289, A038845, A128796.

List([0..30], n-> 4^n*n*(n+1)); # G. C. Greubel, May 10 2019

[(n^2+n)*4^n: n in [0..30]]; // Vincenzo Librandi, Feb 28 2013

I:=[0,8,96]; [n le 3 select I[n] else 12*Self(n-1)-48*Self(n-2)+64*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 28 2013

Table[(n^2 + n)*4^n, {n, 0, 30}] (* Vincenzo Librandi, Feb 28 2013 *)
LinearRecurrence[{12,-48,64},{0,8,96},30] (* Harvey P. Dale, Feb 27 2015 *)

a(n)=(n^2+n)*4^n \\ Charles R Greathouse IV, Feb 28 2013

[4^n*n*(n+1) for n in (0..30)] # G. C. Greubel, May 10 2019

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 8*x/(1-4*x)^3.
a(n) = 8*A038845(n-1). (End)
a(n) = 12*a(n-1) -48*a(n-2) +64*a(n-3). - Vincenzo Librandi, Feb 28 2013
E.g.f.: 8*x*(1 + 2*x)*exp(4*x). - G. C. Greubel, May 10 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 3*log(4/3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(5/4) - 1. (End)

A128798 n(n-1)4^n.

Original entry on oeis.org

0, 0, 32, 384, 3072, 20480, 122880, 688128, 3670016, 18874368, 94371840, 461373440, 2214592512, 10468982784, 48855252992, 225485783040, 1030792151040, 4672924418048, 21028159881216, 94008244174848, 417814418554880

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 (12,-48,64).

Cf. A036289, A007758.

[(n^2-n)*4^n: n in [0..20]]; // Vincenzo Librandi, Feb 10 2013

CoefficientList[Series[32 x^2/(1 - 4 x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2013 *)
LinearRecurrence[{12,-48,64},{0,0,32},30] (* Harvey P. Dale, Dec 23 2015 *)

G.f.: 32*x^2/(1 - 4*x)^3. - Vincenzo Librandi, Feb 10 2013

a(n) =32*A038845(n-2). - R. J. Mathar, Apr 26 2015

cp's OEIS Frontend

A111636 Triangle read by rows: T(n,k) (0<=k<=n) is the number of labeled graphs having k blue nodes and n-k green ones and only nodes of different colors can be joined by an edge.

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A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

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A045724 Convolution of Catalan numbers A000108 with A020918.

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A367022 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = 4^n * hypergeom([1/2, -n - 1, -n], [2, 2], x).

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A129532 3n(n-1)4^(n-2).

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A082150 A transform of C(n,2).

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A052780 Expansion of e.g.f. x^2*exp(4*x).

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A052780 Expansion of e.g.f. x^2exp(4x).

A128798 n(n-1)4^n.