A002105 -id:A002105 - cp's OEIS frontend

A117513 Number of ways of arranging 2*n tokens in a row, with 2 copies of each token from 1 through n, such that between every pair of tokens labeled i (i = 1..n-1) there is exactly one taken labeled i+1.

1, 2, 12, 136, 2480, 66336, 2446528, 118984832, 7378078464, 568142287360, 53189920492544, 5949749335001088, 783686338494312448, 120058889459865165824, 21166245289132322242560, 4254864627502524070395904, 967406173145278971994898432, 247007221085479721384365129728

Offset: 1

Nan Zang (nzang(AT)cs.ucsd.edu), Apr 28 2006

nonn

From Paul Barry, Oct 12 2009: (Start)
The aerated sequence is (2^(n/2 - 1) + 0^(n/2)/2)*((1 + (-1)^n)/2)*n!*[x^n](1 + x*tan(x/2)).
Multiples of the unsigned Genocchi numbers A110501: (1, 1, 3, 17, 155,...)*(1, 2, 4, 8, 16,...). (End)

Michael De Vlieger, Table of n, a(n) for n = 1..258
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

Cf. A002105, A110501, A117514, A117515.

a := n -> (-2)^n*(1 - 2^(2*n))*bernoulli(2*n);
seq(a(n), n = 1..18); # Peter Luschny, Jul 26 2021

Array[(-2)^#*(1 - 2^(2 #))*BernoulliB[2 #] &, 18] (* Michael De Vlieger, Jul 26 2021 *)

# Algorithm of L. Seidel (1877)
# n -> [a(1), ..., a(n)] for n >= 1.
def A117513_list(n) :
    D = [0]*(n+2); D[1] = 1
    R = []; z = 1/2; b = True
    for i in(0..2*n-1) :
        h = i//2 + 1
        if b :
            for k in range(h-1, 0, -1) : D[k] += D[k+1]
            z *= 2
        else :
            for k in range(1, h+1, 1) :  D[k] += D[k-1]
        b = not b
        if b : R.append(D[h]*z)
    return R
A117513_list(15) # Peter Luschny, Jun 29 2012

G.f.: 1/(1-2*x/(1-4*x/(1-8*x/(1-12*x/(1-18*x/(1-24*x/(1-32*x/(1-.../(1-2* floor((n+2)^2/4)*x/(1-... (continued fraction). - Paul Barry, Dec 03 2009
G.f.: T(0), where T(k) = 1 - x*(2*k+2)*(k+1)/( x*(2*k+2)*(k+1) - 1/( 1 - x*(2*k+2)*(k+2)/( x*(2*k+2)*(k+2) - 1/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013
a(n) = (-2)^n*(1 - 2^(2*n))*Bernoulli(2*n). - Peter Luschny, Jul 26 2021

More terms from Paul Barry, Oct 12 2009

A273352 a(n) = 2^(2n+2) F(n) where F(n) is Ramanujan's F(n) = Sum_{k>=1} k^(4n-1)/(e^(Pik)-1) - 16^n Sum_{k>=1} k^(4n-1)/(e^(4Pik)-1).

Original entry on oeis.org

1, 34, 11056, 14873104, 56814228736, 495812444583424, 8575634961418940416, 265929039218907754399744, 13722623393637762299131396096, 1112372064432735526930220874072064, 135292015985218004848567636630910795776, 23782283324940089109797537284278352042000384

Offset: 1

Marko Riedel, May 20 2016

nonn

Bisection of the reduced tangent numbers, A002105. This follows from the formulas. - Franklin T. Adams-Watters, May 22 2016

Math.Stackexchange.Com, Marko R. Riedel et al., Closed form of a sum by Ramanujan

Cf. A002105.

Cf. A000182 (m=2), A293951 (m=3), this seq (m=4), A318258 (m=5).

S := proc(n, k) option remember;
if k=0 then `if`(n=0, 1, 0) else S(n, k-1) + S(n-1, n-k) fi end:
A273352 := n -> S(4*n-1, 4*n-1)/2^(2*n-1):
seq(A273352(n), n=1..12); # Peter Luschny, Jan 18 2017

Table[2^(2*n + 2)*BernoulliB[4*n]*(1 - 2^(4*n))/(8*n), {n, 1, 10}] (* G. C. Greubel, May 21 2016 *)
(* Function LMLlist defined in A293951 *)
LMLlist[4, 13] (* Peter Luschny, Aug 26 2018 *)

a(n) = 2^{2*n+2} * Bernoulli(4*n) * (1-2^(4*n))/(8*n).

A018893 Blasius sequence: from coefficients in expansion of solution to Blasius's equation in boundary layer theory.

Original entry on oeis.org

1, 1, 11, 375, 27897, 3817137, 865874115, 303083960103, 155172279680289, 111431990979621729, 108511603921116483579, 139360142400556127213655, 230624017175131841824732233, 482197541715276031774659298833

Offset: 0

Stan Richardson (stan(AT)maths.ed.ac.uk)

nonn

Number of increasing trilabeled unordered trees. - Markus Kuba, Nov 18 2014

			A(x) = 1 + 1/6*x^3 + 11/720*x^6 + 25/24192*x^9 + 9299/159667200*x^12 + ...
G.f. = 1 + x + 11*x^3 + 375*x^4 + 27897*x^5 + 3817137*x^6 + ...

H. T. Davis: Introduction to Nonlinear Differential and Integral Equations (Dover 1962), page 403.

Vaclav Kotesovec, Table of n, a(n) for n = 0..180
Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Inaugural Dissertation, Georg-August-Universität Göttingen, Leipzig, 1907; see p. 8 (a(6) = c_6 and a(7) = c_7 are wrong in the dissertation) [USA access only].
Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. u. Physik 56 (1908), 1-37; see p. 8 (a(6) = c_6 has been corrected, while a(7) = c_7 was re-calculated incorrectly!).
Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. u. Physik 56 (1908), 1-37 [English translation by J. Vanier on behalf of the National Advisory Committee for Aeronautics (NACA), 1950]; see p. 8 (a(6) = c_6 has been corrected, while a(7) = c_7 was re-calculated incorrectly!).
Steven R. Finch, Prandtl-Blasius Flow. [Cached copy, with permission of the author]
W. H. Hager, Blasius: A life in research and education, Exp. Fluids 34(5) (2003), 566-571.
Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.
Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, Discrete Mathematics 339(1) (2016), 227-254.
Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).
Hans Salié, Über die Koeffizienten der Blasiusschen Reihen, Math. Nachr. 14 (1955), 241-248 (1956). [He generalizes the Blasius numbers.]
Wikipedia, Paul Richard Heinrich Blasius.

Cf. A002105, A256522.

a[0] = 1; a[k_] := a[k] = Sum[Binomial[3*k-1, 3*j]*a[j]*a[k-j-1], {j, 0, k-1}]; Table[a[k], {k, 0, 13}] (* Jean-François Alcover, Oct 28 2014 *)

E.g.f. A(x) satisfies (d^3/dx^3)log(A(x)) = A(x). - Vladeta Jovovic, Oct 24 2003
Lim_{n->infinity} (a(n)/(3*n+2)!)^(1/n) = 0.03269425181024... . - Vaclav Kotesovec, Oct 28 2014
T(z) = log(A(z)) satisfies T'''(z)=exp(T(z)), such that F(z)=T'(z) satisfies a Blasius type equation: F'''(z)-F(z)*F''(z)=0. - Markus Kuba, Nov 18 2014
a(n) = Sum_{v = 0..n-1} binomial(3*n-1, 3*v) * a(v) * a(n-1-v) for n >= 1 with a(0) = 1 (Blasius' recurrence). - Petros Hadjicostas, Aug 01 2019

Corrected and extended by Vladeta Jovovic, Oct 24 2003

A093049 n-1 minus exponent of 2 in n, a(0) = 0.

Original entry on oeis.org

0, 0, 0, 2, 1, 4, 4, 6, 4, 8, 8, 10, 9, 12, 12, 14, 11, 16, 16, 18, 17, 20, 20, 22, 20, 24, 24, 26, 25, 28, 28, 30, 26, 32, 32, 34, 33, 36, 36, 38, 36, 40, 40, 42, 41, 44, 44, 46, 43, 48, 48, 50, 49, 52, 52, 54, 52, 56, 56, 58, 57, 60, 60, 62, 57, 64, 64, 66, 65, 68

Offset: 0

Ralf Stephan, Mar 16 2004

nonn

			G.f. = 2*x^3 + x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 4*x^8 + 8*x^9 + 8*x^10 + ... - _Michael Somos_, Jan 25 2020

a(n) = n - A007814(n) - 1 = A093048(n) - 1, n>0.

a(n) is the exponent of 2 in A001761(n+1), A002105(n), A002682(n-1), A006963(n), A036770(n-1), A059837(n), A084623(n), |A003707(n)|, |A011859(n)|.

a[ n_] := If[ n == 0, 0, n - 1 - IntegerExponent[n, 2]]; (* Michael Somos, Jan 25 2020 *)

a(n)=if(n<1,0,if(n%2==0,a(n/2)+n/2-1,n-1))

{a(n) = if( n, n - 1 - valuation(n, 2))}; /* Michael Somos, Jan 25 2020 */

def A093049(n): return n-1-(~n& n-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Recurrence: a(2n) = a(n) + n - 1, a(2n+1) = 2n.

G.f.: sum(k>=0, t^3(t+2)/(1-t^2)^2, t=x^2^k).

A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

Original entry on oeis.org

1, 1, 0, 5, 1, 0, 61, 28, 1, 0, 1385, 1011, 123, 1, 0, 50521, 50666, 11706, 506, 1, 0, 2702765, 3448901, 1212146, 118546, 2041, 1, 0, 199360981, 308869464, 147485535, 24226000, 1130235, 8184, 1, 0

Offset: 0

Peter Luschny, Sep 20 2017

The generalized Eulerian polynomials F_{m}(x) are defined F_{m; 0}(x) = 1 for all m >= 0 and for n > 0:
F_{0; n}(x) = Sum_{k=0..n} A097805(n, k)*(x-1)^(n-k) with coeffs. in A129186.
F_{1; n}(x) = Sum_{k=0..n} A131689(n, k)*(x-1)^(n-k) with coeffs. in A173018.
F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) with coeffs. in A292604.
F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) with coeffs. in A292605.
F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) with coeffs. in A292606.
The case m = 1 are the Eulerian polynomials whose coefficients are the Eulerian numbers which are displayed in Euler's triangle A173018.
Evaluated at x in {-1, 1, 0} these families of polynomials give for the first few m:
F_{m} :   F_{0}    F_{1}    F_{2}    F_{3}    F_{4}
x = -1:  A165326  A155585  A002105  A292609  A292607
x =  1:  A000012  A000142  A000680  A014606  A014608  ... (m*n)!/m!^n
x =  0:    --     A000012  A000364  A002115  A211212  ... m-alternating permutations of length m*n.
Note that the constant terms of the polynomials are the generalized Euler numbers as defined in A181985. In this sense generalized Euler numbers are also generalized Eulerian numbers.

			Triangle starts:
[n\k][    0        1        2       3     4  5  6]
--------------------------------------------------
[0][      1]
[1][      1,       0]
[2][      5,       1,       0]
[3][     61,      28,       1,      0]
[4][   1385,    1011,     123,      1,    0]
[5][  50521,   50666,   11706,    506,    1, 0]
[6][2702765, 3448901, 1212146, 118546, 2041, 1, 0]

G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.

F_{0} = A129186, F_{1} = A173018, F_{2} is this triangle, F_{3} = A292605, F_{4} = A292606.
First column: A000364. Row sums: A000680. Alternating row sums: A002105.
Cf. A181985, A241171.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292604_row := proc(n) if n = 0 then return [1] fi;
add(A241171(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292604_row(n) od;

T[n_, k_] /; 1 <= k <= n := T[n, k] = k (2 k - 1) T[n - 1, k - 1] + k^2 T[n - 1, k]; T[, 1] = 1; T[, _] = 0;
F[2, 0][] = 1; F[2, n][x_] := Sum[T[n, k] (x - 1)^(n - k), {k, 0, n}];
row[n_] := If[n == 0, {1}, Append[CoefficientList[ F[2, n][x], x], 0]];
Table[row[n], {n, 0, 7}] (* Jean-François Alcover, Jul 06 2019 *)

def A292604_row(n):
    if n == 0: return [1]
    S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..6): print(A292604_row(n))

F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) for n>0 and F_{2; 0}(x) = 1.

A322231 E.g.f.: C(x,k) = 1 + Integral S(x,k)D(x,k)^2 dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2S(x,k)^2 = 1, as a triangle of coefficients read by rows.

Original entry on oeis.org

1, 1, 0, 1, 8, 0, 1, 88, 136, 0, 1, 816, 6240, 3968, 0, 1, 7376, 195216, 513536, 176896, 0, 1, 66424, 5352544, 39572864, 51880064, 11184128, 0, 1, 597864, 139127640, 2458228480, 8258202240, 6453433344, 951878656, 0, 1, 5380832, 3535586112, 137220256000, 994697838080, 1889844670464, 978593947648, 104932671488, 0, 1, 48427552, 88992306208, 7233820923904, 102950036177920, 398800479698944, 485265505927168, 178568645312512, 14544442556416, 0

Offset: 0

Paul D. Hanna, Dec 14 2018

Equals a row reversal of triangle A325222.
Compare to cn(x,k) = 1 - Integral sn(x,k)*dn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions (see triangle A060627).
Compare also to Michael Pawellek's generalized elliptic functions.

			E.g.f.: C(x,k) = 1 + x^2/2! + (8*k^2 + 1)*x^4/4! + (136*k^4 + 88*k^2 + 1)*x^6/6! + (3968*k^6 + 6240*k^4 + 816*k^2 + 1)*x^8/8! + (176896*k^8 + 513536*k^6 + 195216*k^4 + 7376*k^2 + 1)*x^10/10! + (11184128*k^10 + 51880064*k^8 + 39572864*k^6 + 5352544*k^4 + 66424*k^2 + 1)*x^12/12! + (951878656*k^12 + 6453433344*k^10 + 8258202240*k^8 + 2458228480*k^6 + 139127640*k^4 + 597864*k^2 + 1)*x^14/14! + ...
such that C(x,k)^2 - S(x,k)^2 = 1.
This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. C(x,k) begins:
1;
1, 0;
1, 8, 0;
1, 88, 136, 0;
1, 816, 6240, 3968, 0;
1, 7376, 195216, 513536, 176896, 0;
1, 66424, 5352544, 39572864, 51880064, 11184128, 0;
1, 597864, 139127640, 2458228480, 8258202240, 6453433344, 951878656, 0;
1, 5380832, 3535586112, 137220256000, 994697838080, 1889844670464, 978593947648, 104932671488, 0;
1, 48427552, 88992306208, 7233820923904, 102950036177920, 398800479698944, 485265505927168, 178568645312512, 14544442556416, 0; ...
RELATED SERIES.
The related series S(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
S(x,k) = x + (2*k^2 + 1)*x^3/3! + (16*k^4 + 28*k^2 + 1)*x^5/5! + (272*k^6 + 1032*k^4 + 270*k^2 + 1)*x^7/7! + (7936*k^8 + 52736*k^6 + 36096*k^4 + 2456*k^2 + 1)*x^9/9! + (353792*k^10 + 3646208*k^8 + 4766048*k^6 + 1035088*k^4 + 22138*k^2 + 1)*x^11/11! + (22368256*k^12 + 330545664*k^10 + 704357760*k^8 + 319830400*k^6 + 27426960*k^4 + 199284*k^2 + 1)*x^13/13! + ...
The related series D(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
D(x,k) = 1 + k^2*x^2/2! + (5*k^4 + 4*k^2)*x^4/4! + (61*k^6 + 148*k^4 + 16*k^2)*x^6/6! + (1385*k^8 + 6744*k^6 + 2832*k^4 + 64*k^2)*x^8/8! + (50521*k^10 + 410456*k^8 + 383856*k^6 + 47936*k^4 + 256*k^2)*x^10/10! + (2702765*k^12 + 32947964*k^10 + 54480944*k^8 + 17142784*k^6 + 780544*k^4 + 1024*k^2)*x^12/12! + (199360981*k^14 + 3402510924*k^12 + 8760740640*k^10 + 5199585280*k^8 + 686711040*k^6 + 12555264*k^4 + 4096*k^2)*x^14/14! + ...

Paul D. Hanna, Table of n, a(n) for n = 0..860 terms of this triangle read by rows 0..40.

Cf. A322230 (S), A322232 (D), A001818 (row sums), A002105.

Cf. A325222 (row reversal).

N=10;
{S=x;C=1;D=1; for(i=1,2*N, S = intformal(C*D^2 +O(x^(2*N+1))); C = 1 + intformal(S*D^2); D = 1 + k^2*intformal(S*C*D));}
for(n=0,N, for(j=0,n, print1( (2*n)!*polcoeff(polcoeff(C,2*n,x),2*j,k),", ")) ;print(""))

E.g.f. C = C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n) * k^(2*j) / (2*n)!, along with related series S = S(x,k) and D = D(x,k), satisfies:
(1a) S = Integral C*D^2 dx.
(1b) C = 1 + Integral S*D^2 dx.
(1c) D = 1 + k^2 * Integral S*C*D dx.
(2a) C^2 - S^2 = 1.
(2b) D^2 - k^2*S^2 = 1.
(3a) C + S = exp( Integral D^2 dx ).
(3b) D + k*S = exp( k * Integral C*D dx ).
(4a) S = sinh( Integral D^2 dx ).
(4b) S = sinh( k * Integral C*D dx ) / k.
(4c) C = cosh( Integral D^2 dx ).
(4d) D = cosh( k * Integral C*D dx ).
(5a) d/dx S = C*D^2.
(5b) d/dx C = S*D^2.
(5c) d/dx D = k^2 * S*C*D.
From Paul D. Hanna, Mar 31 2019, Apr 20 2019 (Start):
Given sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions, with i^2 = -1, k' = sqrt(1-k^2), then
(6a) S = -i * sn( i * Integral D dx, k),
(6b) C = cn( i * Integral D dx, k),
(6c) D = dn( i * Integral D dx, k).
(7a) S = sc( Integral D dx, k') = sn(Integral D dx, k')/cn(Integral D dx, k'),
(7b) C = nc( Integral D dx, k') = 1/cn(Integral D dx, k'),
(7c) D = dc( Integral D dx, k') = dn(Integral D dx, k')/cn(Integral D dx, k'). (End)
Row sums equal ( (2*n)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.
Diagonal T(n+1,n) = 2^n*A002105(n+1), for n>=0, where A002105 gives the reduced tangent numbers.

A327000 A(n, k) = A309522(n, k) - A327001(n, k) for n >= 0 and k >= 3, square array read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 6, 3, 9, 26, 10, 117, 68, 100, 35, 2574, 4500, 517, 365, 126, 70005, 748616, 199155, 4163, 1302, 462, 2082759, 192426260, 282846568, 10499643, 36180, 4606, 1716, 65061234, 59688349943, 799156187475, 141482705378, 663488532, 341733, 16284

Offset: 0

Peter Luschny, Aug 12 2019

			Array starts:
n\k [  3    4        5            6                 7 ]
[0]    1,   6,       26,          100,              365, ...            [A125107]
[1]    1,   9,       68,          517,              4163, ...           [A048742]
[2]    3,   117,     4500,        199155,           10499643, ...       [A326995]
[3]    10,  2574,    748616,      282846568,        141482705378, ...   [A327002]
[4]    35,  70005,   192426260,   799156187475,     4961959681629275, ...
[5]    126, 2082759, 59688349943, 3097220486457142, 278271624962638244163, ...
   A001700,

Cf. A327001, A309522, A125107, A048742, A001700, A291973, A326999, A326995, A327002.

ListTools:-Flatten([seq(seq(A309522(n-k, k) - A327001(n-k, k), k=3..n), n=3..10)]);

The columns for k = 0, 1, 2 are suppressed as they are identical 0.
A(0, k) = A000108(k) - A011782(k).
A(1, k) = A000142(k) - A000110(k).
A(2, k) = A002105(k) - A005046(k-1) for k >= 1.
A(3, k) = A018893(k) - A291973(k).
A(4, k) = A326999(k) - A291975(k).

A117514 Number of ways of arranging 2n tokens in a row, with 2 copies of each token from 1 through n, such that the first token is a 1 and between every pair of tokens labeled i (i=1..n) there is exactly one taken labeled (i+1 mod n).

Original entry on oeis.org

1, 1, 2, 18, 248, 5560, 174752

Offset: 1

Nan Zang (nzang(AT)cs.ucsd.edu), Apr 28 2006

nonn

Cf. A002105, A117513, A117515.

A117515 Number of ways of arranging 2n tokens in a row, with 2 copies of each token from 1 through n, such that between every pair of tokens labeled i (i=1..n) there is exactly one taken labeled (i+1 mod n).

Original entry on oeis.org

1, 2, 6, 72, 1240, 33360

Offset: 1

Nan Zang (nzang(AT)cs.ucsd.edu), Apr 28 2006

nonn

Cf. A002105, A117514, A117513.

A154604 Hankel transform of reduced tangent numbers.

Original entry on oeis.org

1, 1, 3, 54, 9720, 26244000, 1488034800000, 2362404048480000000, 135019896025206528000000000, 347259290825980971841536000000000000, 49121618545275670528799969525760000000000000000

Offset: 0

Paul Barry, Jan 12 2009

Hankel transform of A002105 (with interpolated zeros).

Hankel transform of A154603.

G. C. Greubel, Table of n, a(n) for n = 0..34
Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2. - _N. J. A. Sloane_, Dec 27 2012

Cf. A002105, A154603.

[n eq 0 select 1 else (&*[(Binomial(k+1,2))^(n-k+1): k in [1..n]]): n in [0..15]]; // G. C. Greubel, May 30 2024

Table[Product[(k*(k+1)/2)^(n - k + 1), {k, 1, n}], {n, 0, 12}] (* Vaclav Kotesovec, Nov 13 2022 *)

a(n) = prod(k=1, n, binomial(k+1,2)^(n-k+1)); \\ Michel Marcus, Nov 13 2022

[product((binomial(k+1,2))^(n-k+1) for k in range(1,n+1)) for n in range(16)] # G. C. Greubel, May 30 2024

a(n) = Product_{k=1..n} C(k+1,2)^(n-k+1).

a(n) ~ n^(n^2 + 3*n + 7/3) * Pi^(n + 3/2) / (A^2 * 2^((n^2 - n - 3)/2) * exp(3*n^2/2 + 3*n - 1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 13 2022

cp's OEIS Frontend

A117513 Number of ways of arranging 2*n tokens in a row, with 2 copies of each token from 1 through n, such that between every pair of tokens labeled i (i = 1..n-1) there is exactly one taken labeled i+1.

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A273352 a(n) = 2^(2n+2) F(n) where F(n) is Ramanujan's F(n) = Sum_{k>=1} k^(4n-1)/(e^(Pi*k)-1) - 16^n* Sum_{k>=1} k^(4n-1)/(e^(4*Pi*k)-1).

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A018893 Blasius sequence: from coefficients in expansion of solution to Blasius's equation in boundary layer theory.

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A093049 n-1 minus exponent of 2 in n, a(0) = 0.

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A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

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A322231 E.g.f.: C(x,k) = 1 + Integral S(x,k)*D(x,k)^2 dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2*S(x,k)^2 = 1, as a triangle of coefficients read by rows.

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A327000 A(n, k) = A309522(n, k) - A327001(n, k) for n >= 0 and k >= 3, square array read by ascending antidiagonals.

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A273352 a(n) = 2^(2n+2) F(n) where F(n) is Ramanujan's F(n) = Sum_{k>=1} k^(4n-1)/(e^(Pik)-1) - 16^n Sum_{k>=1} k^(4n-1)/(e^(4Pik)-1).

A322231 E.g.f.: C(x,k) = 1 + Integral S(x,k)D(x,k)^2 dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2S(x,k)^2 = 1, as a triangle of coefficients read by rows.