A005046 -id:A005046 - cp's OEIS frontend

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.

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)

A260876 Number of m-shape set partitions, square array read by ascending antidiagonals, A(m,n) for m, n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 5, 5, 1, 1, 11, 31, 15, 7, 1, 1, 36, 365, 379, 52, 11, 1, 1, 127, 6271, 25323, 6556, 203, 15, 1, 1, 463, 129130, 3086331, 3068521, 150349, 877, 22, 1, 1, 1717, 2877421, 512251515, 3309362716, 583027547, 4373461, 4140, 30

Offset: 0

Peter Luschny, Aug 02 2015

A set partition of m-shape is a partition of a set with cardinality m*n for some n >= 0 such that the sizes of the blocks are m times the parts of the integer partitions of n.
If m = 0, all possible sizes are zero. Thus the number of set partitions of 0-shape is the number of integer partitions of n (partition numbers A000041).
If m = 1, the set is {1, 2, ..., n} and the set of all possible sizes are the integer partitions of n. Thus the number of set partitions of 1-shape is the number of set partitions (Bell numbers A000110).
If m = 2, the set is {1, 2, ..., 2n} and the number of set partitions of 2-shape is the number of set partitions into even blocks A005046.
From Petros Hadjicostas, Aug 06 2019: (Start)
Irwin (1916) proved the following combinatorial result: Assume r_1, r_2, ..., r_n are positive integers and we have r_1*r_2*...*r_n objects. We divide them into r_1 classes of r_2*r_3*...*r_n objects each, then each class into r_2 subclasses of r_3*...*r_n objects each, and so on. We call each such classification, without reference to order, a "classification" par excellence. He proved that the total number of classifications is (r_1*r_2*...*r_n)!/( r1! * (r_2!)^(r_1) * (r_3!)^(r_1*r_2) * ... (r_n!)^(r_1*r_2*...*r_{n-1}) ).
Apparently, this problem appeared in Carmichael's "Theory of Numbers".
This result can definitely be used to prove some special cases of my conjecture below. (End)

			[ n ] [0  1   2       3        4           5              6]
[ m ] ------------------------------------------------------
[ 0 ] [1, 1,  2,      3,       5,          7,            11]  A000041
[ 1 ] [1, 1,  2,      5,      15,         52,           203]  A000110
[ 2 ] [1, 1,  4,     31,     379,       6556,        150349]  A005046
[ 3 ] [1, 1, 11,    365,   25323,    3068521,     583027547]  A291973
[ 4 ] [1, 1, 36,   6271, 3086331, 3309362716, 6626013560301]  A291975
        A260878,A309725, ...
For example the number of set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] is 1, 84 and 280 respectively. Thus A(3,3) = 365.
Formatted as a triangle:
[1]
[1, 1]
[1, 1,   2]
[1, 1,   2,    3]
[1, 1,   4,    5,     5]
[1, 1,  11,   31,    15,    7]
[1, 1,  36,  365,   379,   52,  11]
[1, 1, 127, 6271, 25323, 6556, 203, 15]
.
From _Peter Luschny_, Aug 14 2019: (Start)
For example consider the case n = 4. There are five integer partitions of 4:
  P = [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]. The shapes are m times the parts of the integer partitions: S(m) = [[4m], [3m, m], [2m, 2m], [2m, m, m], [m, m, m, m]].
* In the case m = 1 we look at set partitions of {1, 2, 3, 4} with sizes in  [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]] which gives rise to [1, 4, 3, 6, 1] with sum 15.
* In the case m = 2 we look at set partitions of {1, 2, .., 8} with sizes in [[8], [6, 2], [4, 4], [4, 2, 2], [2, 2, 2, 2]] which gives rise to [1, 28, 35, 210, 105] with sum 379.
* In the case m = 0 we look at set partitions of {} with sizes in [[0], [0, 0], [0, 0], [0, 0, 0], [0, 0, 0, 0]] which gives rise to [1, 1, 1, 1, 1] with sum 5 (because the only partition of the empty set is the set that contains the empty set, thus from the definition T(0,4) = Sum_{S(0)} card({0}) = A000041(4) = 5).
If n runs through 0, 1, 2,... then the result is an irregular triangle in which the n-th row lists multinomials for partitions of [m*n] which have only parts which are multiples of m. These are the triangles A080575 (m = 1), A257490 (m = 2), A327003 (m = 3), A327004 (m = 4). In the case m = 0 the triangle is A000012 subdivided into rows of length A000041. See the cross references how this case integrates into the full picture.
(End)

Alois P. Heinz, Antidiagonals n = 0..54, flattened
Frank Irwin, Solution to Problem 223 proposed by T. E. Mason in October 1914, Amer. Math. Monthly 23(9) (1916), 352-353.

-----------------------------------------------------------------
[m] |  multi-   |  sum of   |  main     |  by       |  comple-  |
    |  nomials  |  rows     |  diagonal |  size     |  mentary  |
-----------------------------------------------------------------
[0] |  A000012  |  A000041  |  A000012  |  A008284  |  A081362  |
[1] |  A036040  |  A000110  |  A000012  |  A048993  |  A000587  |
[2] |  A257490  |  A005046  |  A001147  |  A156289  |  A260884  |
[3] |  A327003  |  A291973  |  A025035  |  A291451  |  A291974  |
[4] |  A327004  |  A291975  |  A025036  |  A291452  |  A291976  |
Cf. A326996 (main diagonal), A260883 (ordered), A260875 (complementary).
Columns include A000012, A260878, A309725.

A:= proc(m, n) option remember; `if`(m=0, combinat[numbpart](n),
      `if`(n=0, 1, add(binomial(m*n-1, m*k-1)*A(m, n-k), k=1..n)))
    end:
seq(seq(A(d-n, n), n=0..d), d=0..10);  # Alois P. Heinz, Aug 14 2019

A[m_, n_] := A[m, n] = If[m==0, PartitionsP[n], If[n==0, 1, Sum[Binomial[m n - 1, m k - 1] A[m, n - k], {k, 1, n}]]];
Table[Table[A[d - n, n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

def A260876(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return sum(SetPartitions(sum(s), s).cardinality() for s in shapes)
for m in (0..4): print([A260876(m,n) for n in (0..6)])

From Petros Hadjicostas, Aug 02 2019: (Start)
A(m, 2) = 1 + (1/2) * binomial(2*m, m) for m >= 1.
A(m, 3) = 1 + binomial(3*m, m) + (3*m)!/(6 * (m!)^3) for m >= 1.
A(m, 4) = (1/4!) * multinomial(4*m, [m, m, m, m]) + (1/2) * multinomial(4*m, [2*m, m, m]) + multinomial(4*m, [m, 3*m]) + (1/2) * multinomial(4*m, [2*m, 2*m]) + 1 for m >= 1.
Conjecture: For n >= 0, let P be the set of all possible lists (a_1,...,a_n) of nonnegative integers such that a_1*1 + a_2*2 + ... + a_n*n = n. Consider terms of the form multinomial(n*m, m*[1,..., 1,2,..., 2,..., n,..., n])/(a_1! * a_2! * ... * a_n!), where in the list [1,...,1,2,...,2,...,n,...,n] the number 1 occurs a_1 times, 2 occurs a_2 times, ..., and n occurs a_n times. (Here a_n = 0 or 1.) Summing these terms over P we get A(m, n) provided m >= 1. (End)
Conjecture for a recurrence: A(m, n) = Sum_{k = 0..n-1} binomial(m*n - 1, m*k) * A(m, k) with A(m, 0) = 1 for m >= 1 and n >= 0. (Unfortunately, the recurrence does not hold for m = 0.) - Petros Hadjicostas, Aug 12 2019

A291451 Triangle read by rows, expansion of e.g.f. exp(x(exp(z)/3 + 2exp(-z/2)* cos(z*sqrt(3)/2)/3 - 1)), nonzero coefficients of z.

Original entry on oeis.org

1, 0, 1, 0, 1, 10, 0, 1, 84, 280, 0, 1, 682, 9240, 15400, 0, 1, 5460, 260260, 1401400, 1401400, 0, 1, 43690, 7128576, 99379280, 285885600, 190590400, 0, 1, 349524, 193360720, 6600492080, 42549306800, 76045569600, 36212176000

Offset: 0

Peter Luschny, Sep 07 2017

			Triangle starts:
[1]
[0, 1]
[0, 1,    10]
[0, 1,    84,     280]
[0, 1,   682,    9240,    15400]
[0, 1,  5460,  260260,  1401400,   1401400]
[0, 1, 43690, 7128576, 99379280, 285885600, 190590400]

Cf. A048993 (m=1), A156289 (m=2), this seq. (m=3), A291452 (m=4).
Diagonal: A000012 (m=1), A001147 (m=2), A025035 (m=3), A025036 (m=4).
Row sums: A000110 (m=1), A005046 (m=2), A291973 (m=3), A291975 (m=4).
Alternating row sums: A000587 (m=1), A260884 (m=2), A291974 (m=3), A291976 (m=4).

CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
A291451_row := proc(n) exp(x*(exp(z)/3+2*exp(-z/2)*cos(z*sqrt(3)/2)/3-1)):
series(%, z, 66): CL((3*n)!*coeff(series(%,z,3*(n+1)),z,3*n),x) end:
for n from 0 to 7 do A291451_row(n) od;
# Alternative:
A291451row := proc(n) local P; P := proc(m, n) option remember;
if n = 0 then 1 else add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
CL(P(3, n), x); seq(%[k+1]/k!, k=0..n) end: # Peter Luschny, Sep 03 2018

P[m_, n_] := P[m, n] = If[n == 0, 1, Sum[Binomial[m*n, m*k]*P[m, n - k]*x, {k, 1, n}]];
row[n_] := Module[{cl = CoefficientList[P[3, n], x]}, Table[cl[[k + 1]]/k!, {k, 0, n}]];
Table[row[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jul 23 2019, after Peter Luschny *)

A291452 Triangle read by rows, expansion of e.g.f. exp(x*(cos(z) + cosh(z) - 2)/2), nonzero coefficients of z.

Original entry on oeis.org

1, 0, 1, 0, 1, 35, 0, 1, 495, 5775, 0, 1, 8255, 450450, 2627625, 0, 1, 130815, 35586525, 727476750, 2546168625, 0, 1, 2098175, 2941884000, 181262956875, 1932541986375, 4509264634875

Offset: 0

Peter Luschny, Sep 07 2017

			Triangle starts:
[1]
[0, 1]
[0, 1,      35]
[0, 1,     495,       5775]
[0, 1,    8255,     450450,      2627625]
[0, 1,  130815,   35586525,    727476750,    2546168625]
[0, 1, 2098175, 2941884000, 181262956875, 1932541986375, 4509264634875]

Cf. A048993 (m=1), A156289 (m=2), A291451 (m=3), this seq. (m=4).
Diagonal: A000012 (m=1), A001147 (m=2), A025035 (m=3), A025036 (m=4).
Row sums: A000110 (m=1), A005046 (m=2), A291973 (m=3), A291975 (m=4).
Alternating row sums: A000587 (m=1), A260884 (m=2), A291974 (m=3), A291976 (m=4).

CL := (f,x) -> PolynomialTools:-CoefficientList(f,x):
A291452_row := proc(n) exp(x*(cos(z)+cosh(z)-2)/2):
series(%, z, 88): CL((4*n)!*coeff(series(%,z,4*(n+1)),z,4*n),x) end:
for n from 0 to 7 do A291452_row(n) od;
# Alternative:
A291452row := proc(n) local P; P := proc(m, n) option remember;
if n = 0 then 1 else add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
CL(P(4, n), x); seq(%[k+1]/k!, k=0..n) end: # Peter Luschny, Sep 03 2018

P[m_, n_] := P[m, n] = If[n == 0, 1, Sum[Binomial[m*n, m*k]*P[m, n - k]*x, {k, 1, n}]];
row[n_] := Module[{cl = CoefficientList[P[4, n], x]}, Table[cl[[k + 1]]/k!, {k, 0, n}]];
Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Jul 23 2019, after Peter Luschny *)

A306347 Expansion of e.g.f. exp((sin(x) + sinh(x))/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 22, 57, 128, 389, 1904, 9329, 38040, 132147, 542648, 3283633, 20997824, 114657097, 536178880, 2784500161, 19876061312, 153326461311, 1034551839872, 6051063485481, 38079448046208, 312420426154893, 2785055242928768, 22141255520251313

Offset: 0

Ilya Gutkovskiy, May 08 2019

nonn

Number of partitions of n-set into blocks congruent to 1 mod 4.

Seiichi Manyama, Table of n, a(n) for n = 0..608

Cf. A002017, A003724, A005046, A013025, A016813, A035451, A291975, A307978, A307979, A351969.

nmax = 28; CoefficientList[Series[Exp[(Sin[x] + Sinh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 28}]

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sin(x)+sinh(x))/2))) \\ Seiichi Manyama, Mar 17 2022

a(n) = if(n==0, 1, sum(k=0, (n-1)\4, binomial(n-1, 4*k)*a(n-4*k-1))); \\ Seiichi Manyama, Mar 17 2022

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * a(n-4*k-1). - Seiichi Manyama, Mar 17 2022

A307978 Expansion of e.g.f. exp((sinh(x) - sin(x))/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 10, 1, 0, 280, 120, 1, 15400, 17160, 2080, 1401401, 3203200, 1290640, 190623040, 775975201, 712150400, 36321556720, 239000886400, 413465452401, 9339501072000, 91625659447400, 266045692290560, 3216459513124001, 42923384190336000, 193108117771690680

Offset: 0

Ilya Gutkovskiy, May 08 2019

nonn

Number of partitions of n-set into blocks congruent to 3 mod 4.

Seiichi Manyama, Table of n, a(n) for n = 0..608

Cf. A002017, A003724, A004767, A005046, A013369, A035462, A291975, A306347, A307979.

nmax = 29; CoefficientList[Series[Exp[(Sinh[x] - Sin[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{3}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 29}]

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sinh(x)-sin(x))/2))) \\ Seiichi Manyama, Mar 17 2022

a(n) = if(n==0, 1, sum(k=0, (n-3)\4, binomial(n-1, 4*k+2)*a(n-4*k-3))); \\ Seiichi Manyama, Mar 17 2022

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/4)} binomial(n-1,4*k+2) * a(n-4*k-3). - Seiichi Manyama, Mar 17 2022

A327001 Generalized Bell numbers, square array read by ascending antidiagonals, A(n, k) for n, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 4, 5, 8, 1, 1, 11, 31, 15, 16, 1, 1, 36, 365, 379, 52, 32, 1, 1, 127, 6271, 25323, 6556, 203, 64, 1, 1, 463, 129130, 3086331, 3068521, 150349, 877, 128, 1, 1, 1717, 2877421, 512251515, 3309362716, 583027547, 4373461, 4140, 256

Offset: 0

Peter Luschny, Aug 12 2019

			[n\k][0  1   2        3        4           5             6]
[ - ] -----------------------------------------------------
[ 0 ] 1, 1,  2,       4,       8,         16,            32  A011782
[ 1 ] 1, 1,  2,       5,      15,         52,           203  A000110
[ 2 ] 1, 1,  4,      31,     379,       6556,        150349  A005046
[ 3 ] 1, 1, 11,     365,   25323,    3068521,     583027547  A291973
[ 4 ] 1, 1, 36,    6271, 3086331, 3309362716, 6626013560301  A291975
       A260878, A326998,
Formatted as a triangle:
[1]
[1, 1]
[1, 1,   2]
[1, 1,   2,    4]
[1, 1,   4,    5,     8]
[1, 1,  11,   31,    15,    16]
[1, 1,  36,  365,   379,   52,  32]
[1, 1, 127, 6271, 25323, 6556, 203, 64]

A260876 (variant based on shapes).
Rows include: A011782, A000110, A005046, A291973, A291975.
Columns include: A260878, A326998.
Cf. A327000.

A327001 := proc(n, k) option remember; if k = 0 then return 1 fi;
add(binomial(n*k - 1, n*j) * A327001(n, j), j = 0..k-1) end:
for n from 0 to 6 do seq(A327001(n, k), k=0..6) od; # row-wise

A[n_, k_] := A[n, k] = If[k == 0, 1, Sum[Binomial[n*k-1, n*j]*A[n, j], {j, 0, k-1}]];
Table[A[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 27 2022 *)

A(n, k) = Sum_{j=0..k-1} binomial(n*k - 1, n*j) * A(n, j) for k > 0, A(n, 0) = 1.

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cp's OEIS Frontend

A081557 Binomial transform of expansion of exp(cosh(x)), A005046.

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A326995 a(n) = A002105(n+1) - A005046(n), reduced tangent numbers minus the number of partitions of a 2*n-set into even blocks.

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Maple

Formula

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

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Maple

Mathematica

Maxima

Formula

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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Maple

Mathematica

Formula

A260876 Number of m-shape set partitions, square array read by ascending antidiagonals, A(m,n) for m, n >= 0.

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Maple

Mathematica

SageMath

Formula

A291451 Triangle read by rows, expansion of e.g.f. exp(x*(exp(z)/3 + 2*exp(-z/2)* cos(z*sqrt(3)/2)/3 - 1)), nonzero coefficients of z.

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Maple

Mathematica

A291452 Triangle read by rows, expansion of e.g.f. exp(x*(cos(z) + cosh(z) - 2)/2), nonzero coefficients of z.

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Maple

Mathematica

A291451 Triangle read by rows, expansion of e.g.f. exp(x(exp(z)/3 + 2exp(-z/2)* cos(z*sqrt(3)/2)/3 - 1)), nonzero coefficients of z.