A001761 -id:A001761 - cp's OEIS frontend

A106519 a(n) = (2/n)binomial(2n-2, n-1) - (1/(2n))Sum_{d|n} Moebius(d)binomial(2n/d, n/d).

1, 1, 1, 2, 3, 9, 19, 58, 160, 499, 1527, 4940, 16001, 53187, 178305, 606330, 2079863, 7203864, 25138879, 88367780, 312577245, 1112119079, 3977502767, 14294207172, 51596165898, 186998138529, 680272336906, 2483341820512, 9094756956909

Offset: 1

F. Chapoton, May 30 2005

A simple formula with no known combinatorial interpretation. This should give the multiplicity of the trivial module in some sequence of modules of dimension (2*n-2)!/n! over the symmetric groups S_n induced from modules of dimension (2*n-2)!/n!(n-1)! over the cyclic groups C_n.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000108, A001761, A008683, A098796.

A106519:= func< n | 2*Catalan(n-1) - (1/(2*n))*(&+[Round(Gamma(2*n/d +1)/Gamma(n/d +1)^2)*MoebiusMu(d): d in Divisors(n)]) >;
[A106519(n): n in [1..30]]; // G. C. Greubel, Aug 06 2021

a:= n -> (2/n)*( binomial(2*n-2, n-1) - (1/4)*add(NumberTheory[Moebius](d)*binomial(2*n/d, n/d), d = Divisors(n)) );
seq(a(n), n = 1..30); # modified by G. C. Greubel, Aug 06 2021

f[n_] := Block[{d = Divisors[n]}, 2*Binomial[2n-2, n-1]/n - Plus @@ (MoebiusMu[d]*Binomial[2*n/d, n/d])/(2n)]; Table[f[n], {n, 29}] (* Robert G. Wilson v, May 31 2005 *)

a(n) = (2*binomial(2*n-2, n-1) - sumdiv(n, d, moebius(d)*binomial(2*n/d, n/d))/2)/n; \\ Michel Marcus, Aug 06 2021

def a(n):
    return binomial(2*n-2,n-1)*2//n - sum(moebius(n//d)*binomial(2*d,d) for d in divisors(n))//(2*n) # F. Chapoton, May 31 2020

a(n) = (2/n)*binomial(2*n-2, n-1) - (1/(2*n))*Sum_{d|n} Moebius(d)*binomial(2*n/d, n/d).
a(n) = 2*A000108(n-1) - (1/(2*n))*Sum_{d|n} Moebius(d)*(n/d + 1)*A000108(n/d). - G. C. Greubel, Aug 06 2021
a(prime(n)) = A098796(n). - Peter Bala, Aug 20 2025

More terms from Robert G. Wilson v, May 31 2005

a(1) = 1 prepended by G. C. Greubel, Aug 06 2021

A106520 a(n) = A068875(n-1) - A003239(n).

Original entry on oeis.org

1, 0, 0, 0, 2, 4, 18, 48, 156, 472, 1526, 4852, 16000, 52940, 178276, 605520, 2079862, 7201084, 25138878, 88358520, 312576996, 1112087012, 3977502766, 14294093652, 51596165872, 186997738504, 680272334202, 2483340387644, 9094756956908

Offset: 1

F. Chapoton, May 30 2005

nonn

This is the multiplicity of the trivial module in a sequence of modules of dimension (2*n-2)!/n! over the symmetric groups S_n, induced from modules of dimension (2*n-2)!/(n!*(n-1)!) (Catalan) over the cyclic groups C_n.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Chapoton, On some anticyclic operads, Algebraic and Geometric Topology 5 (2005), paper no. 4, pages 53-69.

Cf. A000010, A000108, A001761.

A106520:= func< n | 2*Catalan(n-1) - (1/(2*n))*(&+[Round(Gamma(2*n/d +1)/Gamma(n/d +1)^2)*EulerPhi(d): d in Divisors(n)]) >;
[A106520(n): n in [1..40]]; // G. C. Greubel, Aug 06 2021

with(numtheory);
a:= proc(n) (2/n)*binomial(2*n-2, n-1) - (1/(2*n))*add(phi(d)*binomial(2*n/d, n/d), d = divisors(n)) end:
seq(a(n), n = 1..40);

a[n_]:= 2/n*Binomial[2*n-2, n-1] - 1/(2*n)*DivisorSum[n, EulerPhi[#]* Binomial[2*n/#, n/#]&]; Table[a[n], {n, 40}] (* Jean-François Alcover, Feb 20 2017 *)

a(n) = (2/n) * binomial(2*n-2, n-1) - 1/(2*n) * sumdiv(n, d, eulerphi(d) * binomial(2*n/d, n/d)); \\ Michel Marcus, Aug 08 2021

def a(n): return 2*catalan_number(n-1) - (1/(2*n))*sum(euler_phi(n/d)*binomial(2*d, d) for d in divisors(n))
[a(n) for n in (1..40)] # G. C. Greubel, Aug 06 2021

a(n) = (2/n) * binomial(2*n-2, n-1) - 1/(2*n) * Sum_{d divides n} phi(d) * binomial(2*n/d, n/d).

a(n) = 2*A000108(n-1) - (1/(2*n))*Sum_{d divides n} (n/d + 1)*A000108(n/d) * A000010(d). - G. C. Greubel, Aug 06 2021

Terms a(1) to a(4) prepended by G. C. Greubel, Aug 06 2021

A114595 Triangle of the numbers of unique-valued sequences of all lengths (from 1 to 2n-1) consisting of unit matrices (="matrix units") of order n.

Original entry on oeis.org

1, 4, 6, 4, 9, 24, 48, 60, 30, 16, 60, 192, 480, 840, 840, 336, 25, 120, 520, 1920, 5700, 12600, 18480, 15120, 5040, 36, 210, 1140, 5520, 22920, 78120, 206640, 393120, 483840, 332640, 95040

Offset: 1

Aleksandar Blazhevski - Cane (CaneB(AT)MT.Net.Mk), Dec 12 2005

First entry of each row is obviously n^2 and the last entries of each row are the terms of A001761.

			T(n,1) = a(n^2 - 2n) = n^2.

A. Blazhevski and P. Sokoloski, Conceptual Introduction to Unique-valued Sequences, Proceedings of the III Congress of Mathematicians of Macedonia (2005), preprint.
Aleksandar Blazhevski - Cane, First 10 rows [Broken link]

Cf. A097635, A001761.

A156653 Triangle T(n,k) = ((-1)^(n+k)/(n+1))Sum_{j=0..n} (-1)^jj!Stirling2(n, j) binomial(n-j, k)*binomial(n+j, j), read by rows.

Original entry on oeis.org

1, 1, 3, 1, 16, 13, 1, 125, 171, 39, 1, 1296, 2551, 1091, 101, 1, 16807, 43653, 28838, 5498, 243, 1, 262144, 850809, 780585, 243790, 24270, 561, 1, 4782969, 18689527, 22278189, 10073955, 1733035, 98661, 1263, 1

Offset: 0

Roger L. Bagula, Feb 12 2009

Row sums are A001761.

			Triangle begins as:
          1;
          1;
          3,         1;
         16,        13,         1;
        125,       171,        39,         1;
       1296,      2551,      1091,       101,         1;
      16807,     43653,     28838,      5498,       243,        1;
     262144,    850809,    780585,    243790,     24270,      561,      1;
    4782969,  18689527,  22278189,  10073955,   1733035,    98661,   1263,    1;
  100000000, 457947691, 677785807, 410994583, 106215619, 10996369, 379693, 2797, 1;

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

Cf. A001761, A008292, A048993.

A156653:= func< n,k | ((-1)^(n+k)/(n+1))*(&+[ (-1)^j*Factorial(j)*StirlingSecond(n, j)*Binomial(n-j, k)*Binomial(n+j, j) : j in [0..n]]) >;
[1] cat [A156653(n,k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Mar 01 2021

T[n_, m_]:= Sum[(-1)^(n+m+k) k! StirlingS2[n, k] Binomial[n-k, m] Binomial[n+k, k], {k, 0, n}]/(n+1);
Prepend[Table[T[n, m], {n,10}, {m, 0, n-1}]//Flatten, 1] (* Peter Luschny, May 11 2020 *)

T(n,m):=sum(k!*stirling2(n,k)*(-1)^(n+m+k)*binomial(n-k,m)*binomial(n+k,k),k,0,n) /(n+1); /* Vladimir Kruchinin, May 11 2020 */

def A156653(n,k): return ((-1)^(n+k)/(n+1))*sum( (-1)^j*factorial(j)* stirling_number2(n, j)*binomial(n-j, k)*binomial(n+j, j) for j in (0..n))
[1]+flatten([[A156653(n,k) for k in (0..n-1)] for n in (1..12)]) # G. C. Greubel, Mar 01 2021

T(n, m) = [x^m] p(x,n) where p(x,n) = (1-x)^(2*n+1)/((n+1)*x^n)*Sum_{k>=0} (k+1)^n* binomial(k, n)*x^k.
T(n, m) = 1/(n+1)*Sum_{k=0..n} (-1)^(n+m+k)*k!*Stirling2(n,k)*C(n-k,m)*C(n+k,k). - Vladimir Kruchinin, May 05 2020
E.g.f. satisfies: A(x,y) = x*E(A(x,y),y), where E(x,y) is e.g.f. of Euler numbers of first kind A008292. - Vladimir Kruchinin, May 05 2020

New name by Vladimir Kruchinin, May 11 2020

A375826 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^(x^2).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 1260, 11088, 99120, 1926720, 32800320, 535328640, 11274642720, 259872088320, 6108539621184, 158608655251200, 4495317057504000, 134114095312404480, 4253953999500357120, 143971794376985272320, 5141239842495675340800

Offset: 0

Seiichi Manyama, Aug 30 2024

nonn

Cf. A001761, A349559.

Cf. A375827.

a(n) = n!*sum(k=0, n\3, (n-2*k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(n/3)} (n-2*k+1)^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.

A048870 Triangle of coefficients of certain Sheffer-polynomials.

Original entry on oeis.org

1, 1, 1, 4, 10, 1, 30, 132, 27, 1, 336, 2232, 696, 52, 1, 5040, 46320, 19500, 2200, 85, 1, 95040, 1141920, 606960, 91800, 5340, 126, 1, 2162160, 32639040, 20991600, 3986640, 310170, 11004, 175, 1, 57657600, 1061746560, 802287360, 183550080

Offset: 0

Wolfdieter Lang

s(n,x) := sum(a(n,m)*x^m,m=0..n) are monic polynomials satisfying s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)=sum(A048786(n,m)*x^m, m=1..n) (row polynomials of triangle A048786) and p(0,x)=1. In the umbral calculus (see reference) the s(n,x) are called Sheffer polynomials for(c(t/(1+4*t)),t/(1+4*t)), where c(x) = g.f. for Catalan numbers A000108. a(n,0) = A001761(n-2) = n!*A000108(n).

S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

A046527, A048786, A000108, A001761.

a(n, m) = (n!/m!)*A046527(n, m) = (n!/m!)*binomial(n, m-1)*(4^(n-m+1)-binomial(2*n, n)/binomial(2*(m-1), m-1))/2, n >= m >= 0, a(n, m) := 0, n

A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)tanh(1)^k) / (2^nn! * (1-tanh(1))^n).

Original entry on oeis.org

0, 1, 0, 1, -1, -1, -1, 0, 0, 3, 1, -5, 18, -13, -7, -11, 70, -135, 65, -10, 45, 111, -609, 1215, -1350, 1275, -621, -141, -1009, 6188, -16758, 27335, -26845, 12474, -2548, 1883, 10977, -81353, 270004, -511791, 584710, -420287, 216468, -70169, -3599, -146691, 1248210, -4715217, 10303461, -14439411

Offset: 0

Gerald McGarvey, Mar 02 2005

For n>0 the row sums = (-1)^(n-1) * (n-1)! For n odd, the sum of the absolute values of the coefficients in the n-th row = (2*(n-1))!/n! (every other entry of A001761).
The sum of the (2n)th du Bois-Reymond constants = 1/5 or is very close to 1/5.
For the 6th and 9th rows, the coefficients were adjusted from results of the residue evaluations so that double factorials ((2n)!! = 2^n*n! (A000165)) are in the denominators. For the 6th row they were multiplied by 3, for the 9th row they were multiplied by 9.
For n>1, Sum_{k=0..n} (n-k+1)*a(n,k) = (-1)^(n)*A001286(n-1) [A001286 are Lah numbers: (n-1)*n!/2].

Eric Weisstein's World of Mathematics, Double Factorial.
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A000142, A000165, A001761, A062545, A062546, A085466, A085467.

Table[2 Residue[x^2/((1+x^2)^n (Tan[x]-x)), {x, I}], {n, 0, 9}]

For n>1, C_2n = -3 - 2 * Residue_{x=i} (x^2/((1+x^2)^n * (tan(x) - x))) (see MathWorld article).

For n>1, Sum_{k=0..n} (-1)^(n+k)*a(n, k) = (2*(n-1))!/n! (i.e., A001761(n-1)).

Added the keyword tabl Gerald McGarvey, Aug 20 2009

A335748 T(n,k) = (-1)^n(binomial(2k,k)/(k+1))Sum_{j=0..n} (-1)^jbinomial(k,j)*j^n. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -1, 4, 0, 1, -12, 30, 0, -1, 28, -180, 336, 0, 1, -60, 750, -3360, 5040, 0, -1, 124, -2700, 21840, -75600, 95040, 0, 1, -252, 9030, -117600, 705600, -1995840, 2162160, 0, -1, 508, -28980, 571536, -5292000, 25280640, -60540480, 57657600

Offset: 0

Peter Luschny, Jul 09 2020

			                             [0] 1
                           [1] 0, 1
                         [2] 0, -1, 4
                       [3] 0, 1, -12, 30
                   [4] 0, -1, 28, -180, 336
                [5] 0, 1, -60, 750, -3360, 5040
          [6] 0, -1, 124, -2700, 21840, -75600, 95040
   [7] 0, 1, -252, 9030, -117600, 705600, -1995840, 2162160
[8] 0, -1, 508, -28980, 571536, -5292000, 25280640, -60540480, 57657600

Cf. A006531 (row sums), A052895 (absolute row sums), T(n,n) = A001761(n) (signed A292220(n)).

T(n, k) = (-1)^n*CatalanNumber(k)*Sum_{j=0..n}(-1)^j*binomial(k, j)*j^n.

A376350 E.g.f. satisfies A(x) = 1/(1 - x^2A(x)^2)^(xA(x)).

Original entry on oeis.org

1, 0, 0, 6, 0, 60, 2520, 1680, 181440, 6138720, 18295200, 1444988160, 46443196800, 357015859200, 25016537145600, 818965321574400, 12259854032025600, 815066633667686400, 28461465853402982400, 691667282863484928000, 45198900807076912896000, 1739192274792359202816000, 60318174486002275287244800

Offset: 0

Seiichi Manyama, Sep 21 2024

nonn

Index entries for reversions of series

Cf. A001761, A184949, A371147.

Cf. A353226.

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^2)^x)/x))

a(n) = n!*sum(k=0, n\2, (n+1)^(n-2*k-1)*abs(stirling(k, n-2*k, 1))/k!);

E.g.f.: (1/x) * Series_Reversion( x*(1 - x^2)^x ).

a(n) = n! * Sum_{k=0..floor(n/2)} (n+1)^(n-2*k-1) * |Stirling1(k,n-2*k)|/k!.

Previous Showing 31-39 of 39 results.

cp's OEIS Frontend

A106519 a(n) = (2/n)*binomial(2*n-2, n-1) - (1/(2*n))*Sum_{d|n} Moebius(d)*binomial(2*n/d, n/d).

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A106520 a(n) = A068875(n-1) - A003239(n).

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A114595 Triangle of the numbers of unique-valued sequences of all lengths (from 1 to 2n-1) consisting of unit matrices (="matrix units") of order n.

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A156653 Triangle T(n,k) = ((-1)^(n+k)/(n+1))*Sum_{j=0..n} (-1)^j*j!*Stirling2(n, j)* binomial(n-j, k)*binomial(n+j, j), read by rows.

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A375826 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^(x^2).

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A048870 Triangle of coefficients of certain Sheffer-polynomials.

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A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)*tanh(1)^k) / (2^n*n! * (1-tanh(1))^n).

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A106519 a(n) = (2/n)binomial(2n-2, n-1) - (1/(2n))Sum_{d|n} Moebius(d)binomial(2n/d, n/d).

A156653 Triangle T(n,k) = ((-1)^(n+k)/(n+1))Sum_{j=0..n} (-1)^jj!Stirling2(n, j) binomial(n-j, k)*binomial(n+j, j), read by rows.

A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)tanh(1)^k) / (2^nn! * (1-tanh(1))^n).

A335748 T(n,k) = (-1)^n(binomial(2k,k)/(k+1))Sum_{j=0..n} (-1)^jbinomial(k,j)*j^n. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.

A376350 E.g.f. satisfies A(x) = 1/(1 - x^2A(x)^2)^(xA(x)).