A052714 -id:A052714 - cp's OEIS frontend

A052720 Expansion of e.g.f.: (1 - 4x + 3x^2)(1 - 2x - sqrt(1-4x))/2 - x^2 + 2x^3.

0, 0, 0, 0, 0, 0, 720, 30240, 1088640, 39916800, 1556755200, 65383718400, 2964061900800, 144815595724800, 7602818775552000, 427447366714368000, 25646842002862080000, 1636734826000834560000, 110752389892723138560000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 676

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052721, A052722, A052723.

Cf. A000108, A003517.

spec := [S,{B=Union(Z,C),C=Prod(B,B),S=Prod(C,C,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[If[n<6, 0, 6*(n-2)!*Binomial[n-4, 2]*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052720(n):
    if (n<6): return 0
    else: return 6*factorial(n-2)*binomial(n-4,2)*catalan_number(n-3)
[A052720(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(1)=0; a(2)=0; a(4)=0; a(3)=0; a(5)=0; a(6)=720; a(n+3) = (10+8*n)*a(n+2) + (22-27*n-19*n^2)*a(n+1) - (60-66*n+6*n^2+12*n^3)*a(n).
a(n) = n!*A003517(n-4). - R. J. Mathar, Oct 18 2013
From G. C. Greubel, May 28 2022: (Start)
G.f.: 6!*x^6*Hypergeometric2F0([3, 7/2], [], 4*x).
E.g.f.: (1/2)*(1 - 6*x + 9*x^2 - 2*x^3 - (1 - 4*x + 3*x^2)*sqrt(1-4*x)). (End)

A052721 Expansion of e.g.f. x(1-2x)(1 - 2x - sqrt(1-4*x))/2 - x^3.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 2880, 70560, 1935360, 59875200, 2075673600, 79913433600, 3387499315200, 156883562035200, 7884404656128000, 427447366714368000, 24869664972472320000, 1545805113445232640000, 102232975285590589440000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 677

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052722, A052723.

Cf. A000108, A002057.

spec := [S,{C=Union(B,Z),B=Prod(C,C),S=Prod(B,B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[If[n<5, 0, 2*n*(n-2)!*(n-4)*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052721(n):
    if (n<5): return 0
    else: return 2*n*factorial(n-2)*(n-4)*catalan_number(n-3)
[A052721(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=120, a(6)=2880, (n+2)*a(n+2) = (6*n^2 + 8*n - 8)*a(n+1) + (40 + 44*n = 4*n^2 - 8*n^3)*a(n).
a(n) = 2*Pi^(-1/2)*4^(n-3)*Gamma(n-5/2)*n*(n-4) for n>3.  - Mark van Hoeij, Oct 30 2011
a(n) = n!*A002057(n-5). - R. J. Mathar, Oct 18 2013
From G. C. Greubel, May 28 2022: (Start)
G.f.: 4!*x*(d/dx)( x^5 * Hypergeometric2F0([2, 5/2], [], 4*x) ).
E.g.f.: (x/2)*(1 - 4*x + 2*x^2 - (1-2*x)*sqrt(1-4*x)). (End)

A052722 Expansion of e.g.f. (1 - 2x - sqrt(1-4x))^2 * (1 - sqrt(1-4*x))/8.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 3600, 100800, 3024000, 99792000, 3632428800, 145297152000, 6351561216000, 301699157760000, 15487223431680000, 854894733428736000, 50516506975334400000, 3182539939446067200000, 212985365178313728000000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 678

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052723.

Cf. A000108, A000344.

spec := [S,{C=Union(B,Z),B=Prod(C,C),S=Prod(B,B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[((1-2x-Sqrt[1-4x])^2 (1-Sqrt[1-4x]))/8,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 30 2021 *)
Table[If[n<5, 0, 10*(n-2)!*Binomial[n-3,2]*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052722(n):
    if (n<5): return 0
    else: return 10*factorial(n-2)*binomial(n-3,2)*catalan_number(n-3)
[A052722(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(0) = a(1) = a(2) = a(3) = a(4) = 0, a(5)=120, a(n+3) = (9+7*n)*a(n+2) + (14 - 19*n - 13*n^2)*a(n+1) - (20 + 22*n - 2*n^2 - 4*n^3)*a(n).
a(n) = n!*A000344(n-3). - R. J. Mathar, Oct 18 2013
From G. C. Greubel, May 28 2022: (Start)
G.f.: 5!*x^5*hypergeometric2F0([5/2, 3], [], 4*x).
E.g.f.: (1/2)*(1 - 5*x + 5*x^2 - (1 - 3*x + x^2)*sqrt(1-4*x)). (End)

A052734 a(n) = 4^(n-1) * n! * Catalan(n-1) for n > 0, with a(0) = 0.

Original entry on oeis.org

0, 1, 8, 192, 7680, 430080, 30965760, 2724986880, 283398635520, 34007836262400, 4625065731686400, 703009991216332800, 118105678524343910400, 21731444848479279513600, 4346288969695855902720000, 938798417454304874987520000, 217801232849398730997104640000, 54014705746650885287281950720000, 14259882317115833715842434990080000

Offset: 0

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

For n>0, the number of fully-parenthesized expressions that you can form with n operands and 4 types of binary operators. - Yogy Namara (yogy.namara(AT)gmail.com), Mar 07 2010

a(n+1) is the number of square roots of any permutation in S_{16*n} whose disjoint cycle decomposition consists of 2*n cycles of length 8. - Luis Manuel Rivera Martínez, Feb 26 2015

			Let's say the 4 types of binary operators are +, -, *, and /. Then, with 3 operands {a, b, c}, we can form expressions such as ((b+a)/c), (a-(c-b)), (c*(b+a)), etc. There are a(3)=192 such expressions. - Yogy Namara (yogy.namara(AT)gmail.com), Mar 07 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 690.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).
W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012.

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), A052714 (or A144828) (m=2), A221954 (m=3), this sequence (m=4), A221953 (m=5), A221955 (m=6).

Equal to A000108 if all operands and all operators are indistinguishable.

[0] cat [Catalan(n-1)*4^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

spec := [S,{B=Prod(C,C),S=Union(B,Z),C=Union(B,S,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq((2*n)!/n! * 4^n, n = 0..10);

Join[{0}, Table[CatalanNumber[n-1] 4^(n-1) n!, {n, 1, 20}]] (* Vincenzo Librandi, Mar 11 2013 *)

[0]+[4^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

E.g.f.: (1 - sqrt(1-16*x))/8.
Recurrence: a(1)=1, 8*(1 - 2*n)*a(n) + a(n+1) = 0.
a(n) = 16^n*Gamma(n+1/2)/sqrt(Pi).
a(0) = 0, a(1) = 1; a(n) = 4 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 09 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 + e^(1/16)*sqrt(Pi)*erf(1/4)/4, where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/16)*sqrt(Pi)*erfi(1/4)/4, where erfi is the imaginary error function. (End)

Entry revised by N. J. A. Sloane, Feb 04 2013 and Feb 06 2013

A221954 a(n) = 3^(n-1) * n! * Catalan(n-1).

Original entry on oeis.org

1, 6, 108, 3240, 136080, 7348320, 484989120, 37829151360, 3404623622400, 347271609484800, 39588963481267200, 4988209398639667200, 688372897012274073600, 103255934551841111040000, 16727461397398259988480000, 2910578283147297237995520000, 541367560665397286267166720000, 107190777011748662680899010560000, 22510063172467219162988792217600000

Offset: 1

N. J. A. Sloane, Feb 03 2013

nonn

a(n+1) is the number of square roots of any permutation in S_{12*n} whose disjoint cycle decomposition consists of 2*n cycles of length 6. - Luis Manuel Rivera Martínez, Feb 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..200
W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), A052714 (or A144828) (m=2), this sequence (m=3), A052734 (m=4), A221953 (m=5), A221955 (m=6).

Cf. A000108.

[Catalan(n-1)*3^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

A221954:= n-> (3^(n-1)*n!/(2*(2*n-1))*binomial(2*n,n); seq(A221954(n), n=1..30); # G. C. Greubel, Apr 02 2021

Table[CatalanNumber[n-1] 3^(n-1) n!, {n, 20}] (* Vincenzo Librandi, Mar 11 2013 *)

my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-12*x))/6)) \\ Michel Marcus, Mar 04 2015

[3^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

a(n) = 6*(2*n-3)*a(n-1) with a(1)=1. - Bruno Berselli, Mar 11 2013
E.g.f.: (1 - sqrt(1-12*x))/6. - Luis Manuel Rivera Martínez, Mar 04 2015
a(n) = 12^(n-1) * Gamma(n - 1/2) / sqrt(Pi). - Daniel Suteu, Jan 06 2017
a(1) = 1; a(n) = 3 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 09 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 + e^(1/12)*sqrt(Pi)*erf(1/(2*sqrt(3)))/(2*sqrt(3)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/12)*sqrt(Pi)*erfi(1/(2*sqrt(3)))/(2*sqrt(3)), where erfi is the imaginary error function. (End)

A221953 a(n) = 5^(n-1) * n! * Catalan(n-1).

Original entry on oeis.org

1, 10, 300, 15000, 1050000, 94500000, 10395000000, 1351350000000, 202702500000000, 34459425000000000, 6547290750000000000, 1374931057500000000000, 316234143225000000000000, 79058535806250000000000000, 21345804667687500000000000000, 6190283353629375000000000000000, 1918987839625106250000000000000000

Offset: 1

N. J. A. Sloane, Feb 03 2013

nonn

a(n+1) is the number of square roots of any permutation in S_{20*n} whose disjoint cycle decomposition consists of 2*n cycles of length 10. - Luis Manuel Rivera Martínez, Feb 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..200
W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), A052714 (or A144828) (m=2), A221954 (m=3), A052734 (m=4), this sequence (m=5), A221955 (m=6).

Cf. A000108.

[Catalan(n-1)*5^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

A221953:= n-> (5^(n-1)*n!/(2*(2*n-1))*binomial(2*n,n); seq(A221953(n), n=1..30); # G. C. Greubel, Apr 02 2021

Table[CatalanNumber[n - 1]  5^(n-1)  n!, {n, 20}] (* Vincenzo Librandi, Mar 11 2013 *)

my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-20*x))/10)) \\ Michel Marcus, Mar 04 2015

[5^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

a(n) = 10*(2*n-3)*a(n-1) with a(1)=1. - Bruno Berselli, Mar 11 2013
E.g.f.: (1 - sqrt(1-20*x))/10. - Luis Manuel Rivera Martínez, Mar 04 2015
a(1) = 1; a(n) = 5 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 10 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 + e^(1/20)*sqrt(Pi)*erf(1/(2*sqrt(5)))/(2*sqrt(5)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/20)*sqrt(Pi)*erfi(1/(2*sqrt(5)))/(2*sqrt(5)), where erfi is the imaginary error function. (End)

A221955 a(n) = 6^(n-1) * n! * Catalan(n-1).

Original entry on oeis.org

1, 12, 432, 25920, 2177280, 235146240, 31039303680, 4842131374080, 871583647334400, 177803064056217600, 40539098604817612800, 10215852848414038425600, 2819575386162274605465600, 845872615848682381639680000, 274062727534973091651256320000, 95373829182170635894637199360000, 35479064455767476552805038161920000

Offset: 1

N. J. A. Sloane, Feb 03 2013

nonn

a(n+1) is the number of square roots of any permutation in S_{24*n} whose disjoint cycle decomposition consists of 2*n cycles of length 12. - Luis Manuel Rivera Martínez, Feb 28 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..200
W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), A052714 (or A144828) (m=2), A221954 (m=3), A052734 (m=4), A221953 (m=5), this sequence (m=6).

Cf. A000108.

[Catalan(n-1)*6^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

A221955:= n-> 3*6^(n-2)*n!*binomial(2*n,n)/(2*n-1); seq(A221955(n), n=1..30); # G. C. Greubel, Apr 02 2021

Table[CatalanNumber[n-1] 6^(n-1) n!, {n, 20}] (* Vincenzo Librandi, Mar 11 2013 *)
nxt[{n_,a_}]:={n+1,12a(2n-1)}; NestList[nxt,{1,1},20][[;;,2]] (* Harvey P. Dale, Sep 21 2024 *)

my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-24*x))/12)) \\ Michel Marcus, Mar 04 2015

[6^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

a(n) = 12*(2*n-3)*a(n-1) with a(1)=1. - Bruno Berselli, Mar 11 2013
E.g.f.: (1-sqrt(1-24*x))/12. - Luis Manuel Rivera Martínez, Mar 04 2015
a(1) = 1; a(n) = 6 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 10 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 + e^(1/24)*sqrt(Pi)*erf(1/(2*sqrt(6)))/(2*sqrt(6)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/24)*sqrt(Pi)*erfi(1/(2*sqrt(6)))/(2*sqrt(6)), where erfi is the imaginary error function. (End)

A174376 Triangle T(n, k) = n!q^k/(n-k)! if floor(n/2) > k-1 otherwise n!q^(n-k)/k!, with q = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 48, 8, 1, 1, 10, 80, 80, 10, 1, 1, 12, 120, 960, 120, 12, 1, 1, 14, 168, 1680, 1680, 168, 14, 1, 1, 16, 224, 2688, 26880, 2688, 224, 16, 1, 1, 18, 288, 4032, 48384, 48384, 4032, 288, 18, 1, 1, 20, 360, 5760, 80640, 967680, 80640, 5760, 360, 20, 1

Offset: 0

Roger L. Bagula, Mar 17 2010

Row sums are: {1, 2, 6, 14, 66, 182, 1226, 3726, 32738, 105446, 1141242, ...}.

			Triangle begins as:
  1;
  1,  1;
  1,  4,   1;
  1,  6,   6,    1;
  1,  8,  48,    8,     1;
  1, 10,  80,   80,    10,      1;
  1, 12, 120,  960,   120,     12,     1;
  1, 14, 168, 1680,  1680,    168,    14,    1;
  1, 16, 224, 2688, 26880,   2688,   224,   16,   1;
  1, 18, 288, 4032, 48384,  48384,  4032,  288,  18,  1;
  1, 20, 360, 5760, 80640, 967680, 80640, 5760, 360, 20,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A159623 (q=1), this sequence (q=2), A174377 (q=3), A174378 (q=4).

Cf. A052714.

T[n_, k_, q_]:= If[Floor[n/2]>=k, n!*q^k/(n-k)!, n!*q^(n-k)/k!];
Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten

f=factorial
def T(n,k,q): return f(n)*q^k/f(n-k) if ((n//2)>k-1) else f(n)*q^(n-k)/f(k)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 28 2021

T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 2.
T(n, n-k) = T(n, k).
T(2*n, n) = A052714(n+1). - G. C. Greubel, Nov 28 2021

Edited by G. C. Greubel, Nov 28 2021

A147626 Octo-factorial numbers (5).

Original entry on oeis.org

1, 6, 84, 1848, 55440, 2106720, 96909120, 5233092480, 324451733760, 22711621363200, 1771506466329600, 152349556104345600, 14320858273808486400, 1460727543928465612800, 160680029832131217408000, 18960243520191483654144000, 2388990683544126940422144000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..325

Cf. A034908, A045755, A048994, A049210, A051189, A052714, A132393, A147625.

[n le 1 select 1 else (8*n-10)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022

s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,5,2*5!,8}];lst
Table[8^(n-1)*Pochhammer[3/4, n-1], {n,40}] (* G. C. Greubel, Oct 21 2022 *)

[8^(n-1)*rising_factorial(3/4, n-1) for n in range(1,40)] # G. C. Greubel, Oct 21 2022

a(n+1) = Sum_{k=0..n} A132393(n,k)*6^k*8^(n-k). - Philippe Deléham, Nov 09 2008
a(n) = (-2)^n*Sum_{k=0..n} 4^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 2*x/G(0), where G(k) = 1 + 1/(1 - 2*x*(8*k+6)/(2*x*(8*k+6) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
From G. C. Greubel, Oct 21 2022: (Start)
a(n) = 8^n * Pochhammer(n, 3/4) = -2^(3*n-1) * Pochhammer(n, -1/4).
a(n) = (8*n - 10)*a(n-1). (End)
Sum_{n>=1} 1/a(n) = 1 + (e/8^2)^(1/8)*(Gamma(3/4) - Gamma(3/4, 1/8)). - Amiram Eldar, Dec 20 2022

A164961 Triangle read by rows: T[n,m] = quadruple factorials A001813(n) * binomials A007318(n,m).

Original entry on oeis.org

1, 2, 2, 12, 24, 12, 120, 360, 360, 120, 1680, 6720, 10080, 6720, 1680, 30240, 151200, 302400, 302400, 151200, 30240, 665280, 3991680, 9979200, 13305600, 9979200, 3991680, 665280, 17297280, 121080960, 363242880, 605404800, 605404800

Offset: 0

Tilman Neumann, Sep 02 2009

Row sums give A052714. - Tilman Neumann, Sep 07 2009

Triangle T(n,k), read by rows, given by (2, 4, 6, 8, 10, 12, 14, ...) DELTA (2, 4, 6, 8, 10, 12, 14, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 07 2012

			Triangle begins:
  1
  2, 2
  12, 24, 12
  120, 360, 360, 120
  1680, 6720, 10080, 6720, 1680

More terms. - _Tilman Neumann_, Sep 07 2009

Cf. A001813, A007318, A052714 (row sums), A084938, A085881.

T(n,k) = A085881(n,k)*2^n. - Philippe Deléham, Jan 07 2012
Recurrence equation: T(n+1,k) = (4*n+2)*(T(n,k) + T(n,k-1)). - Peter Bala, Jul 15 2012
E.g.f.: 1/sqrt(1-4*x-4*x*y). - Peter Bala, Jul 15 2012

cp's OEIS Frontend

A052720 Expansion of e.g.f.: (1 - 4*x + 3*x^2)*(1 - 2*x - sqrt(1-4*x))/2 - x^2 + 2*x^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A052721 Expansion of e.g.f. x*(1-2*x)*(1 - 2*x - sqrt(1-4*x))/2 - x^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A052722 Expansion of e.g.f. (1 - 2*x - sqrt(1-4*x))^2 * (1 - sqrt(1-4*x))/8.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A052734 a(n) = 4^(n-1) * n! * Catalan(n-1) for n > 0, with a(0) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A221954 a(n) = 3^(n-1) * n! * Catalan(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A221953 a(n) = 5^(n-1) * n! * Catalan(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A221955 a(n) = 6^(n-1) * n! * Catalan(n-1).

Views

Author

A052720 Expansion of e.g.f.: (1 - 4x + 3x^2)(1 - 2x - sqrt(1-4x))/2 - x^2 + 2x^3.

A052721 Expansion of e.g.f. x(1-2x)(1 - 2x - sqrt(1-4*x))/2 - x^3.

A052722 Expansion of e.g.f. (1 - 2x - sqrt(1-4x))^2 * (1 - sqrt(1-4*x))/8.

A174376 Triangle T(n, k) = n!q^k/(n-k)! if floor(n/2) > k-1 otherwise n!q^(n-k)/k!, with q = 2, read by rows.