A136630 -id:A136630 - cp's OEIS frontend

A136632 a(n) = Sum_{k=0..n} A136630(n,k) * 2^(nk).

1, 2, 16, 520, 66560, 33882144, 69055086592, 564152735105152, 18462508115518554112, 2418626436468567646929408, 1267795674038260517176495570944, 2658560573512321601282555747644737536

Offset: 0

Paul D. Hanna, Jan 14 2008

nonn

			From _Paul D. Hanna_, Nov 25 2009: (Start)
E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 520*x^3/3! + 66560*x^4/4! +...
A(x) = 1 + sinh(2*x) + sinh(4*x)^2/2! + sinh(8*x)^3/3! + sinh(16*x)^4/4! +...+ sinh(2^n*x)^n/n! +...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(sinh(x)):
G(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 12*x^5/5! + 37*x^6/6! +...+ A003724(n)*x^n/n! +... (End)

Robert Israel, Table of n, a(n) for n = 0..57

Cf. A136630, A003724 (row sums of A136630).

Cf. A003724 (exp(sinh x)). [From Paul D. Hanna, Nov 25 2009]

N:= 20: # to get a(0)..a(N)
E:= add(sinh(2^n*x)^n/n!,n=0..N):
S:= series(E,x,N+1):
seq(coeff(S,x,j)*j!,j=0..N); # Robert Israel, Jan 17 2018

{a(n)=sum(k=0,n,2^(n*k)*polcoeff(x^k/prod(j=0,k\2,1-(2*j+k-2*(k\2))^2*x^2 +x*O(x^n)),n))}

{a(n)=n!*polcoeff(sum(k=0,n,sinh(2^k*x +x*O(x^n))^k/k!),n)} \\ Paul D. Hanna, Nov 25 2009

{a(n)=n!*polcoeff(exp(2^n*sinh(x +x*O(x^n))),n)} \\ Paul D. Hanna, Nov 25 2009

From Paul D. Hanna, Nov 25 2009: (Start)
E.g.f.: Sum_{n>=0} sinh(2^n*x)^n/n!.
a(n) = [x^n/n! ] exp(2^n*sinh(x)).
(End)

A136631 Antidiagonal sums of triangle A136630, omitting antidiagonals of all zeros.

Original entry on oeis.org

1, 1, 2, 6, 28, 177, 1449, 14869, 185230, 2738962, 47287352, 939759621, 21241309681, 540698975061, 15370957337418, 484433735633218, 16817886069720724, 639545680226171989, 26507567678760284105

Offset: 0

Paul D. Hanna, Jan 14 2008

nonn

Cf. A136630, A003724 (row sums of A136630).

{a(n)=sum(k=0,n,polcoeff(1/prod(j=0,k\2,1-(2*j+(k%2))^2*x^2 +x*O(x^(2*n-2*k))),2*n-2*k))}

a(n) = Sum_{k=0..n} [x^(2n-2k)] Product_{j=0..[k/2]} 1/(1 - (2j + k-2[k/2])^2*x^2).

A003724 Number of partitions of n-set into odd blocks.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 37, 128, 457, 1872, 8169, 37600, 188685, 990784, 5497741, 32333824, 197920145, 1272660224, 8541537105, 59527313920, 432381471509, 3252626013184, 25340238127989, 204354574172160, 1699894200469849, 14594815769038848, 129076687233903673

Offset: 0

R. H. Hardin

			G.f. = 1 + x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 37*x^6 + 128*x^7 + 457*x^8 + ...

L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225, 2nd line of table.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..592 (first 101 terms from T. D. Noe)
J. Riordan, Letter, Jul 06 1978
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

See A136630 for the table of partitions of an n-set into k odd blocks.
For partitions into even blocks see A005046 and A156289.
Cf. A000009, A000110. A002017, A009623.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*irem(j, 2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2015

a[n_] := Sum[((-1)^i*(k - 2*i)^n*Binomial[k, i])/(2^k*k!), {k, 1, n}, {i, 0, k}]; a[0] = 1; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Dec 21 2011, after Vladimir Kruchinin *)
With[{nn=30},CoefficientList[Series[Exp[Sinh[x]],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Apr 06 2012 *)
Table[Sum[BellY[n, k, Mod[Range[n], 2]], {k, 0, n}], {n, 0, 24}] (* Vladimir Reshetnikov, Nov 09 2016 *)

a(n):=sum(1/2^k*sum((-1)^i*binomial(k,i)*(k-2*i)^n,i,0,k)/k!,k,1,n); /* Vladimir Kruchinin, Aug 22 2010 */

E.g.f.: exp ( sinh x ).
a(n) = sum(1/2^k*sum((-1)^i*C(k,i)*(k-2*i)^n, i=0..k)/k!, k=1..n). - Vladimir Kruchinin, Aug 22 2010
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A002017 and A009623. - Peter Bala, Dec 06 2011
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1). - Ilya Gutkovskiy, Jul 11 2021
O.g.f A(X) satisfies A(x) = 1 + x*( A(x/(1-x))/(1-x) + A(x/(1+x))/(1+x) )/2. - Paul D. Hanna, Aug 19 2024

A005046 Number of partitions of a 2n-set into even blocks.

Original entry on oeis.org

1, 1, 4, 31, 379, 6556, 150349, 4373461, 156297964, 6698486371, 337789490599, 19738202807236, 1319703681935929, 99896787342523081, 8484301665702298804, 802221679220975886631, 83877585692383961052499, 9640193854278691671399436, 1211499609050804749310115589

Offset: 0

N. J. A. Sloane

Conjecture: Taking the sequence modulo an integer k gives an eventually periodic sequence. For example, the sequence taken modulo 10 is [1, 1, 4, 1, 9, 6, 9, 1, 4, 1, 9, 6, 9, 1, 4, 1, 9, 6, 9, ...], with an apparent period [1, 4, 1, 9, 6, 9] beginning at a(1), of length 6. Cf. A006154. - Peter Bala, Apr 12 2023

Louis Comtet, Analyse Combinatoire Tome II, pages 61-62.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 225, 3rd line of table.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.
L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 51 terms from T. D. Noe)
C. Ahmed, P. Martin, and V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015-2019.
Steven R. Finch, Moments of sums, April 23, 2004 [Cached copy, with permission of the author]
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
J. Riordan, Letter, Jul 06 1978
J. Shallit, Letter to N. J. A. Sloane, Jan 13 1976.
Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.

See A156289 for the table of partitions of a 2n-set into k even blocks.
For partitions into odd blocks see A003724 and A136630.
Cf. A000110, A003724.

a:= proc(n) option remember;
      `if`(n=0, 1, add(binomial(2*n-1, 2*k-1) *a(n-k), k=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 12 2011
# second Maple program:
a := n -> add(binomial(2*n,k)*(-1)^k*BellB(k,1/2)*BellB(2*n-k,1/2), k=0..2*n):
seq(a(n), n=0..18); # after Emanuele Munarini,_Peter Luschny_, Sep 10 2017
B := BellMatrix(n -> modp(n, 2), 31): # defined in A264428.
seq(add(k, k in B[2*n + 1]),n=0..15); # Peter Luschny, Aug 13 2019

NestList[ Factor[ D[#, {x, 2}]] &, Exp[ Cosh[x] - 1], 16] /. x -> 0
a[0] = 1; a[n_] := Sum[Sum[(i-k)^(2*n)*Binomial[2*k, i]*(-1)^i, {i, 0, k-1}]/(2^(k-1)*k!), {k, 1, 2*n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 07 2015, after Vladimir Kruchinin *)
Table[Sum[BellY[2 n, k, 1 - Mod[Range[2 n], 2]], {k, 0, 2 n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
With[{nn=40},Abs[Take[CoefficientList[Series[Exp[Cos[x]-1],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]]] (* Harvey P. Dale, Feb 06 2017 *)

a(n):= sum(1/k!*sum(binomial(k,m)/(2^(m-1))*sum(binomial(m,j) *(2*j-m)^(2*n), j,0,m/2)*(-1)^(k-m), m,0,k), k,1,2*n); /* Vladimir Kruchinin, Aug 05 2010 */

a(n):=sum(sum((i-k)^(2*n)*binomial(2*k,i)*(-1)^(i),i,0,k-1)/(2^(k-1)*k!),k,1,2*n); /* Vladimir Kruchinin, Oct 04 2012 */

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def a(n): return 1 if n==0 else sum(binomial(2*n - 1, 2*k - 1)*a(n - k) for k in range(1, n + 1))
print([a(n) for n in range(21)]) # Indranil Ghosh, Sep 11 2017, after Maple program by Alois P. Heinz

E.g.f.: exp(cosh(x) - 1) (or exp(cos(x)-1) ).
a(n) = Sum_{k=1..n} binomial(2*n-1, 2*k-1)*a(n-k). - Vladeta Jovovic, Apr 10 2003
a(n) = sum(1/k!*sum(binomial(k,m)/(2^(m-1))*sum(binomial(m,j)*(2*j-m)^(2*n),j,0,m/2)*(-1)^(k-m),m,0,k),k,1,2*n), n>0. - Vladimir Kruchinin, Aug 05 2010
a(n) = Sum_{k=1..2*n} Sum_{i=0..k-1} ((i-k)^(2*n)*binomial(2*k,i)*(-1)^i)/(2^(k-1)*k!), n>0, a(0)=1. - Vladimir Kruchinin, Oct 04 2012
E.g.f.: E(0)-1, where E(k) = 2 + (cosh(x)-1)/(2*k + 1 - (cosh(x)-1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013
a(n) = Sum_{k=0..2*n} binomial(2*n,k)*(-1)^k*S_k(1/2)*S_{2*n-k}( 1/2), where S_n(x) is the n-th Bell polynomial (or exponential polynomial). - Emanuele Munarini, Sep 10 2017

A385343 Exponential Riordan array (1, arcsin(x)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 9, 0, 10, 0, 1, 0, 0, 64, 0, 20, 0, 1, 0, 225, 0, 259, 0, 35, 0, 1, 0, 0, 2304, 0, 784, 0, 56, 0, 1, 0, 11025, 0, 12916, 0, 1974, 0, 84, 0, 1, 0, 0, 147456, 0, 52480, 0, 4368, 0, 120, 0, 1, 0, 893025, 0, 1057221, 0, 172810, 0, 8778, 0, 165, 0, 1

Offset: 0

Seiichi Manyama, Jun 26 2025

			Triangle starts:
  1;
  0,   1;
  0,   0,  1;
  0,   1,  0,   1;
  0,   0,  4,   0,  1;
  0,   9,  0,  10,  0,  1;
  0,   0, 64,   0, 20,  0, 1;
  0, 225,  0, 259,  0, 35, 0, 1;

Essentialy same as A121408.
Row sums give A006228.
Cf. A091885, A136630, A185951.

T(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(asin(x)^k/k!, n);

E.g.f. of column k (with leading zeros): arcsin(x)^k / k!

T(n,k) = A121408(n,k) for k > 0.

A000557 Expansion of e.g.f. 1/(1 - 2*sinh(x)).

Original entry on oeis.org

1, 2, 8, 50, 416, 4322, 53888, 783890, 13031936, 243733442, 5064992768, 115780447730, 2887222009856, 77998677862562, 2269232452763648, 70734934220015570, 2351893466832306176, 83086463910558199682, 3107896091715557654528, 122711086194279627711410

Offset: 0

N. J. A. Sloane

Inverse binomial transform of A005923. - Vladimir Reshetnikov, Oct 29 2015

Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin's summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Gregory Dresden, On the Brousseau sums Sum_{i=1..n} i^p*Fibonacci(i), arxiv.org:2206.00115 [math.NT], 2022.
Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, Sums related to the Fibonacci sequence, Husson University (2021).
G. Ledin, Jr., On a certain kind of Fibonacci sums, Fib. Quart., 5 (1967), 45-58.
Prabha Sivaraman Nair, A note on alternating weighted sums of Fibonacci numbers, Math. Montisnigri (2024) Vol. LX, 32-49. See p. 38.
Prabha Sivaraman Nair and Rejikumar Karunakaran, On k-Fibonacci Brousseau Sums, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 8.
R. L. Ollerton and A. G. Shannon, A Note on Brousseau's Summation Problem, Fibonacci Quart. 58 (2020), no. 5, 190-199.
Daniele Parisse, On hypersequences of an arbitrary sequence and their weighted sums, Integers (2024) Vol. 24, Art. No. A70. See p. 25.
Eric Weisstein's MathWorld, Polylogarithm.

Cf. A000045, A000556, A005923, A136630, A320352, A358031, A358032.

Cf. A006154, A107403.

A000557 := proc(n) local k,j; add(add((-1)^j*binomial(k,j)*(k-2*j)^n,j=0..k),k=0..n) end: # Peter Luschny, Jul 31 2011

f[n_] := Sum[ k!*StirlingS2[n, k]*Fibonacci[k + 2], {k, 0, n}]; Array[f, 20, 0] (* Robert G. Wilson v, Aug 16 2011 *)
With[{nn=20},CoefficientList[Series[1/(1-2*Sinh[x]),{x,0,nn}],x]Range[ 0,nn]!] (* Harvey P. Dale, Mar 11 2012 *)
Round@Table[(-1)^n (PolyLog[-n, 1-GoldenRatio]-PolyLog[-n, GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

my(x='x+O('x^30)); Vec(serlaplace(1/(1-2*sinh(x)))) \\ Michel Marcus, May 18 2022

E.g.f.: 1/(1-2*sinh(x)). - Vladeta Jovovic, Jul 06 2002
a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^j*binomial(k,j)*(k-2*j)^n. - Peter Luschny, Jul 31 2011
a(n) = Sum_{k=0..n} k!*Stirling2(n, k)*Fibonacci(k+2).
a(n) ~ n! / (sqrt(5) * log((1+sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, May 04 2015
a(n) = (-1)^n*(Li_{-n}(1-phi)-Li_{-n}(phi))/sqrt(5), where Li_n(x) denotes the polylogarithm, phi=(1+sqrt(5))/2. - Vladimir Reshetnikov, Oct 29 2015
a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
Sum_{k=0..n-1} binomial(n,k)*a(k) = A000556(n). - Greg Dresden, Jun 01 2022
a(n) = A000556(n) + A320352(n). - Seiichi Manyama, Oct 26 2022
a(n) = Sum_{k=0..n} 2^k * k! * A136630(n,k). - Seiichi Manyama, Jun 25 2025

More terms from David W. Wilson

A156289 Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.

Original entry on oeis.org

1, 1, 3, 1, 15, 15, 1, 63, 210, 105, 1, 255, 2205, 3150, 945, 1, 1023, 21120, 65835, 51975, 10395, 1, 4095, 195195, 1201200, 1891890, 945945, 135135, 1, 16383, 1777230, 20585565, 58108050, 54864810, 18918900, 2027025, 1, 65535, 16076985

Offset: 1

Hartmut F. W. Hoft, Feb 07 2009

T(n,k) is the number of partitions of a set of size 2*n into k blocks of even size [Comtet]. For partitions into odd sized blocks see A136630.
See A241171 for the triangle of ordered set partitions of the set {1,2,...,2*n} into k even sized blocks. - Peter Bala, Aug 20 2014
This triangle T(n,k) gives the sum over the M_3 multinomials A036040 for the partitions of 2*n with k even parts, for 1 <= k <= n. See the triangle A257490 with sums over the entries with k parts, and the Hartmut F. W. Hoft program. - Wolfdieter Lang, May 13 2015

			The triangle begins
  n\k|..1.....2......3......4......5......6
  =========================================
  .1.|..1
  .2.|..1.....3
  .3.|..1....15.....15
  .4.|..1....63....210....105
  .5.|..1...255...2205...3150....945
  .6.|..1..1023..21120..65835..51975..10395
  ..
T(3,3) = 15. The 15 partitions of the set [6] into three even blocks are:
  (12)(34)(56), (12)(35)(46), (12)(36)(45),
  (13)(24)(56), (13)(25)(46), (13)(26)(45),
  (14)(23)(56), (14)(25)(36), (14)(26)(35),
  (15)(23)(46), (15)(24)(36), (15)(26)(34),
  (16)(23)(45), (16)(24)(35), (16)(25)(34).
Examples of recurrence relation
 T(4,3) = 5*T(3,2) + 9*T(3,3) = 5*15 + 9*15 = 210;
 T(6,5) = 9*T(5,4) + 25*T(5,5) = 9*3150 + 25*945 = 51975.
 T(4,2) = 28 + 35 = 63 (M_3 multinomials A036040 for partitions of 8 with 3 even parts, namely (2,6) and (4^2)). - _Wolfdieter Lang_, May 13 2015

L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pages 225-226.

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
Richard Olu Awonusika, On Jacobi polynomials P_k(alpha, beta) and coefficients c_j^L(alpha, beta) (k >= 0, L = 5,6; 1 <= j <= L; alpha, beta > -1), The Journal of Analysis (2020).
Thomas Browning, Counting Parabolic Double Cosets in Symmetric Groups, arXiv:2010.13256 [math.CO], 2020.
J. Riordan, Letter, Jul 06 1978

Diagonal T(n, n) is A001147, subdiagonal T(n+1, n) is A001880.
2nd column variant T(n, 2)/3, for 2<=n, is A002450.
3rd column variant T(n, 3)/15, for 3<=n, is A002451.
Sum of the n-th row is A005046.
Cf. A241171, A257468, A257490, A096162.

T := proc(n,k) option remember; `if`(k = 0 and n = 0, 1, `if`(n < 0, 0,
(2*k-1)*T(n-1, k-1) + k^2*T(n-1, k))) end:
for n from 1 to 8 do seq(T(n,k), k=1..n) od; # Peter Luschny, Sep 04 2017

T[n_,k_] := Which[n < k, 0, n == 1, 1, True, 2/Factorial2[2 k] Sum[(-1)^(k + j) Binomial[2 k, k + j] j^(2 n), {j, 1, k}]]
(* alternate computation with function triangle[] defined in A257490 *)
a[n_]:=Map[Apply[Plus,#]&,triangle[n],{2}]
(* Hartmut F. W. Hoft, Apr 26 2015 *)

Recursion: T(n,1)=1 for 1<=n; T(n,k)=0 for 1<=n

Generating function for the k-th column of the triangle T(i+k,k):

G(k,x) = Sum_{i>=0} T(i+k,k)*x^i = Product_{j=1..k} (2*j-1)/(1-j^2*x).

Closed form expression: T(n,k) = (2/(k!*2^k))*Sum_{j=1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n).

From Peter Bala, Feb 21 2011: (Start)

GENERATING FUNCTION

E.g.f. (including a constant 1):

(1)... F(x,z) = exp(x*(cosh(z)-1))

= Sum_{n>=0} R(n,x)*z^(2*n)/(2*n)!

= 1 + x*z^2/2! + (x + 3*x^2)*z^4/4! + (x + 15*x^2 + 15*x^3)*z^6/6! + ....

ROW POLYNOMIALS

The row polynomials R(n,x) begin

... R(1,x) = x

... R(2,x) = x + 3*x^2

... R(3,x) = x + 15*x^2 + 15*x^3.

The egf F(x,z) satisfies the partial differential equation

(2)... d^2/dz^2(F) = x*F + x*(2*x+1)*F' + x^2*F'',

where ' denotes differentiation with respect to x. Hence the row polynomials satisfy the recurrence relation

(3)... R(n+1,x) = x*{R(n,x) + (2*x+1)*R'(n,x) + x*R''(n,x)}

with R(0,x) = 1. The recurrence relation for T(n,k) given above follows from this.

(4)... T(n,k) = (2*k-1)!!*A036969(n,k).

(End)

A007289 Expansion of e.g.f. (sin(2x) + cos(x)) / cos(3x).

Original entry on oeis.org

1, 2, 8, 46, 352, 3362, 38528, 515086, 7869952, 135274562, 2583554048, 54276473326, 1243925143552, 30884386347362, 825787662368768, 23657073914466766, 722906928498737152, 23471059057478981762, 806875574817679474688, 29279357851856595135406

Offset: 0

N. J. A. Sloane and Simon Plouffe

nonn

Arises in the enumeration of alternating 3-signed permutations.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Matthew House, Table of n, a(n) for n = 0..406 (terms 0..200 from Vincenzo Librandi)
R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Preprint, 1994. (Annotated scanned copy)
Richard Ehrenborg and Margaret A. Readdy, Sheffer posets and r-signed permutations, Annales des Sciences Mathématiques du Québec, 19 (1995), 173-196.

Row 3 of A349271.
Cf. A006873, A007286, A225109, A000191 (bisection), A000436 (bisection).
Cf. A000111, A136630, A352638.

A007289 := proc(n) local k,j; add(add((-1)^j*binomial(k,j)*(k-2*j)^n*I^(n-k),j=0..k),k=0..n) end: # Peter Luschny, Jul 31 2011

mx = 17; Range[0, mx]! CoefficientList[ Series[ (Sin[2 x] + Cos[x])/Cos[3 x], {x, 0, mx}], x] (* Robert G. Wilson v, Apr 28 2013 *)

my(x='x+O('x^66)); Vec(serlaplace((sin(2*x) + cos(x)) / cos(3*x))) \\ Joerg Arndt, Apr 28 2013

from mpmath import mp, polylog, im
mp.dps = 32; mp.pretty = True
def aperm3(n): return 2*((1-I)/(1+I))^n*(1+add(binomial(n,j)*polylog(-j,I)*3^j for j in (0..n)))
def A007289(n) : return im(aperm3(n))
[int(A007289(n)) for n in (0..17)] # Peter Luschny, Apr 28 2013

E.g.f.: (sin(2*x) + cos(x)) / cos(3*x).
a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^j*binomial(k,j)*(k-2*j)^n*I^(n-k). - Peter Luschny, Jul 31 2011
a(n) = Im(2*((1-I)/(1+I))^n*(1+sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)*3^j))). - Peter Luschny, Apr 28 2013
a(n) ~ n! * 2^(n+1)*3^(n+1/2)/Pi^(n+1). - Vaclav Kotesovec, Jun 15 2013
a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
From Seiichi Manyama, Jun 25 2025: (Start)
E.g.f.: 1/(1 - 2 * sin(x)).
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A136630(n,k), where i is the imaginary unit. (End)

A119467 A masked Pascal triangle.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 6, 0, 1, 0, 5, 0, 10, 0, 1, 1, 0, 15, 0, 15, 0, 1, 0, 7, 0, 35, 0, 21, 0, 1, 1, 0, 28, 0, 70, 0, 28, 0, 1, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 1, 0, 45, 0, 210, 0, 210, 0, 45, 0, 1, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 1, 0, 66, 0, 495, 0, 924

Offset: 0

Paul Barry, May 21 2006

Row sums are A011782. Diagonal sums are F(n+1)*(1+(-1)^n)/2 (aerated version of A001519). Product by Pascal's triangle A007318 is A119468. Schur product of (1/(1-x),x/(1-x)) and (1/(1-x^2),x).
Exponential Riordan array (cosh(x),x). Inverse is (sech(x),x) or A119879. - Paul Barry, May 26 2006
Rows give coefficients of polynomials p_n(x) = Sum_{k=0..n} (k+1 mod 2)*binomial(n,k)*x^(n-k) having e.g.f. exp(x*t)*cosh(t)= 1*(t^0/0!) + x*(t^1/1!) + (1+x^2)*(t^2/2!) + ... - Peter Luschny, Jul 14 2009
Inverse of the coefficient matrix of the Swiss-Knife polynomials in ascending order of x^i (reversed and aerated rows of A153641). - Peter Luschny, Jul 16 2012
Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*... is equal to A136630 but with the first row and column omitted. - Peter Bala, Jul 28 2014
The row polynomials SKv(n,x) = [(x+1)^n + (x-1)^n]/2 , with e.g.f. cosh(t)*exp(xt), are the umbral compositional inverses of the row polynomials of A119879 (basically the Swiss Knife polynomials SK(n,x) of A153641); i.e., umbrally SKv(n,SK(.,x)) = x^n = SK(n,SKv(.,x)). Therefore, this entry's matrix and A119879 are an inverse pair. Both sequences of polynomials are Appell sequences, i.e., d/dx P(n,x) = n * P(n-1,x) and (P(.,x)+y)^n = P(n,x+y). In particular, (SKv(.,0)+x)^n = SKv(n,x), reflecting that the first column has the e.g.f. cosh(t). The raising operator is R = x + tanh(d/dx); i.e., R SKv(n,x) = SKv(n+1,x). The coefficients of this operator are basically the signed and aerated zag numbers A000182, which can be expressed as normalized Bernoulli numbers. The triangle is formed by multiplying the n-th diagonal of the lower triangular Pascal matrix by the Taylor series coefficient a(n) of cosh(x). More relations for this type of triangle and its inverse are given by the formalism of A133314. - Tom Copeland, Sep 05 2015
The signed version of this matrix has the e.g.f. cos(t) e^{xt}, generating Appell polynomials that have only real, simple zeros and whose extrema are maxima above the x-axis and minima below and situated above and below the zeros of the next lower degree polynomial. The bivariate versions appear on p. 27 of Dimitrov and Rusev in conditions for entire functions that are cosine transforms of a class of functions to have only real zeros. - Tom Copeland, May 21 2020
The n-th row of the triangle is obtained by multiplying by 2^(n-1) the elements of the first row of the limit as k approaches infinity of the stochastic matrix P^(2k-1) where P is the stochastic matrix associated with the Ehrenfest model with n balls. The elements of a stochastic matrix P give the probabilities of arriving in a state j given the previous state i. In particular the sum of every row of the matrix must be 1, and so the sum of the terms of the n-th row of this triangle is 2^(n-1). Furthermore, by the properties of Markov chains, we can interpret P^(2k-1) as the (2k-1)-step transition matrix of the Ehrenfest model and its limit exists and it is again a stochastic matrix. The rows of the triangle divided by 2^(n-1) are the even rows (second, fourth, ...) and the odd rows (first, third, ...) of the limit matrix P^(2k-1). - Luca Onnis, Oct 29 2023

			Triangle begins
  1,
  0, 1,
  1, 0,  1,
  0, 3,  0,  1,
  1, 0,  6,  0,   1,
  0, 5,  0, 10,   0,   1,
  1, 0, 15,  0,  15,   0,   1,
  0, 7,  0, 35,   0,  21,   0,  1,
  1, 0, 28,  0,  70,   0,  28,  0,  1,
  0, 9,  0, 84,   0, 126,   0, 36,  0, 1,
  1, 0, 45,  0, 210,   0, 210,  0, 45, 0, 1
p[0](x) = 1
p[1](x) = x
p[2](x) = 1 + x^2
p[3](x) = 3*x + x^3
p[4](x) = 1 + 6*x^2 + x^4
p[5](x) = 5*x + 10*x^3 + x^5
Connection with A136630: With the arrays M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1        \/1        \/1        \      /1         \
|0 1      ||0 1      ||0 1      |      |0 1       |
|1 0 1    ||0 0 1    ||0 0 1    |... = |1 0  1    |
|0 3 0 1  ||0 1 0 1  ||0 0 0 1  |      |0 4  0 1  |
|1 0 6 0 1||0 0 3 0 1||0 0 1 0 1|      |1 0 10 0 1|
|...      ||...      ||...      |      |...       |
- _Peter Bala_, Jul 28 2014

Paul and Tatjana Ehrenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift, vol. 8 (1907), pp. 311-314.

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 28.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Tom Copeland, Skipping over Dimensions, Juggling Zeros in the Matrix, 2020.
D. Dimitrov and P. Rusev, Zeros of entire Fourier transforms, East Journal on Approximations, Vol. 17, No. 1, p. 1-108, 2011.
Miguel Méndez and Rafael Sánchez, On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops, arXiv:1707.00336 [math.CO], 2017, Section 4.3, Example 4.
Miguel A. Méndez and Rafael Sánchez Lamoneda, Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials, The Electronic Journal of Combinatorics 25(3) (2018), #P3.25.
Luca Onnis, Animation of the Ehrenfest model.
Wikipedia, Ehrenfest model.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A131047, A153641, A162590.
From Peter Luschny, Jul 14 2009: (Start)
Cf. A034839, A162590.
p[n](k), n=0,1,...
k= 0: 1,  0,   1,    0,    1,   0, ... A128174
k= 1: 1,  1,   2,    4,    8,  16, ... A011782
k= 2: 1,  2,   5,   14,   41, 122, ... A007051
k= 3: 1,  3,  10,   36,  136,      ... A007582
k= 4: 1,  4,  17,   76,  353,      ... A081186
k= 5: 1,  5,  26,  140,  776,      ... A081187
k= 6: 1,  6,  37,  234, 1513,      ... A081188
k= 7: 1,  7,  50,  364, 2696,      ... A081189
k= 8: 1,  8,  65,  536, 4481,      ... A081190
k= 9: 1,  9,  82,  756, 7048,      ... A060531
k=10: 1, 10, 101, 1030,            ... A081192
p[n](k), k=0,1,...
p[0]: 1,1,1,1,1,1, ....... A000012
p[1]: 0,1,2,3,4,5, ....... A001477
p[2]: 1,2,5,10,17,26, .... A002522
p[3]: 0,4,14,36,76,140, .. A079908 (End)
Cf. A000182, A133314, A153641.

a119467 n k = a119467_tabl !! n !! k
a119467_row n = a119467_tabl !! n
a119467_tabl = map (map (flip div 2)) $
               zipWith (zipWith (+)) a007318_tabl a130595_tabl
-- Reinhard Zumkeller, Mar 23 2014

/* As triangle */ [[Binomial(n, k)*(1 + (-1)^(n - k))/2: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 26 2015

# Polynomials: p_n(x)
p := proc(n,x) local k, pow; pow := (n,k) -> `if`(n=0 and k=0,1,n^k);
add((k+1 mod 2)*binomial(n,k)*pow(x,n-k),k=0..n) end;
# Coefficients: a(n)
seq(print(seq(coeff(i!*coeff(series(exp(x*t)*cosh(t),t,16),t,i),x,n),n=0..i)),i=0..8); # Peter Luschny, Jul 14 2009

Table[Binomial[n, k] (1 + (-1)^(n - k))/2, {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 06 2015 *)
n = 15; "n-th row"
mat = Table[Table[0, {j, 1, n + 1}], {i, 1, n + 1}];
mat[[1, 2]] = 1;
mat[[n + 1, n]] = 1;
For[i = 2, i <= n, i++, mat[[i, i - 1]] = (i - 1)/n ];
For[i = 2, i <= n, i++, mat[[i, i + 1]] = (n - i + 1)/n];
mat // MatrixForm;
P2 = Dot[mat, mat];
R1 = Simplify[
  Eigenvectors[Transpose[P2]][[1]]/
   Total[Eigenvectors[Transpose[P2]][[1]]]]
R2 = Table[Dot[R1, Transpose[mat][[k]]], {k, 1, n + 1}]
odd = R2*2^(n - 1) (* _Luca Onnis *)

@CachedFunction
def A119467_poly(n):
    R = PolynomialRing(ZZ, 'x')
    x = R.gen()
    return R.one() if n==0 else R.sum(binomial(n,k)*x^(n-k) for k in range(0,n+1,2))
def A119467_row(n):
    return list(A119467_poly(n))
for n in (0..10) : print(A119467_row(n)) # Peter Luschny, Jul 16 2012

G.f.: (1-x*y)/(1-2*x*y-x^2+x^2*y^2);
T(n,k) = C(n,k)*(1+(-1)^(n-k))/2;
Column k has g.f. (1/(1-x^2))*(x/(1-x^2))^k*Sum_{j=0..k+1} binomial(k+1,j)*sin((j+1)*Pi/2)^2*x^j.
Column k has e.g.f. cosh(x)*x^k/k!. - Paul Barry, May 26 2006
Let Pascal's triangle, A007318 = P; then this triangle = (1/2) * (P + 1/P). Also A131047 = (1/2) * (P - 1/P). - Gary W. Adamson, Jun 12 2007
Equals A007318 - A131047 since the zeros of the triangle are masks for the terms of A131047. Thus A119467 + A131047 = Pascal's triangle. - Gary W. Adamson, Jun 12 2007
T(n,k) = (A007318(n,k) + A130595(n,k))/2, 0<=k<=n. - Reinhard Zumkeller, Mar 23 2014

Edited by N. J. A. Sloane, Jul 14 2009

A007788 Number of augmented Andre 3-signed permutations: E.g.f. (1-sin(3*x))^(-1/3).

Original entry on oeis.org

1, 1, 4, 19, 136, 1201, 13024, 165619, 2425216, 40132801, 740882944, 15091932019, 336257744896, 8134269015601, 212309523595264, 5946914908771219, 177934946000306176, 5663754614516217601, 191097349696090537984, 6812679868133940475219, 255885704427935576621056

Offset: 0

R. Ehrenborg (ehrenbor(AT)lacim.uqam.ca) and M. A. Readdy (readdy(AT)lacim.uqam.ca)

nonn

It appears that all members are of the form 3k+1. - Ralf Stephan, Nov 12 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Preprint submitted to Ann. Sci. Math. Quebec, 1994. (Annotated scanned copy)
R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Ann. Sci. Math. Québec, 19 (1995), no. 2, 173-196.
R. Ehrenborg and M. A. Readdy, The r-cubical lattice and a generalization of the cd-index, European J. Combin. 17 (1996), no. 8, 709-725.

Cf. A001586, A144015, A230134, A227544, A235128, A230114.
Cf. A007863, A235135, A235132.
Cf. A007559, A136630.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( (1-Sin(3*x))^(-1/3) ))); // G. C. Greubel, Mar 05 2020

m:=20; S:=series( (1-sin(3*x))^(-1/3), x, m+1): seq(j!*coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 05 2020

With[{nn=20},CoefficientList[Series[(1-Sin[3x])^(-1/3),{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Nov 23 2011 *)

Vec(serlaplace( (1-sin(3*x))^(-1/3) +O('x^20) )) \\ G. C. Greubel, Mar 05 2020

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*(3*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

m=20;
def A007788_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1-sin(3*x))^(-1/3) ).list()
a=A007788_list(m+1); [factorial(n)*a[n] for n in (0..m)] # G. C. Greubel, Mar 05 2020

E.g.f.: (1-sin(3*x))^(-1/3).
a(n) ~ n! * 2*6^n/(Pi^(n+2/3)*n^(1/3)*Gamma(2/3)). - Vaclav Kotesovec, Jun 25 2013
a(n) = Sum_{k=0..n} A007559(k) * (3*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

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cp's OEIS Frontend

A136632 a(n) = Sum_{k=0..n} A136630(n,k) * 2^(nk).

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A136631 Antidiagonal sums of triangle A136630, omitting antidiagonals of all zeros.

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A003724 Number of partitions of n-set into odd blocks.

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A005046 Number of partitions of a 2n-set into even blocks.

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A385343 Exponential Riordan array (1, arcsin(x)).

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A000557 Expansion of e.g.f. 1/(1 - 2*sinh(x)).

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A156289 Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.

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A007289 Expansion of e.g.f. (sin(2x) + cos(x)) / cos(3x).