A243665 -id:A243665 - cp's OEIS frontend

A094088 E.g.f. 1/(2-cosh(x)) (even coefficients).

1, 1, 7, 121, 3907, 202741, 15430207, 1619195761, 224061282907, 39531606447181, 8661323866026007, 2307185279184885001, 734307168916191403507, 275199311597682485597221, 119956934012963778952439407

Offset: 0

Ralf Stephan, Apr 30 2004

With alternating signs, e.g.f.: 1/(2-cos(x)).
7 divides a(3n+2). Ira Gessel remarks: For any odd prime p, the coefficients of 1/(2-cosh(x)) as e.g.f. are periodic with period dividing p-1.
Consider the sequence defined by a(0) = 1; thereafter a(n) = c*Sum_{k = 1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
a(n) is the number of ordered set partitions of {1,2,...,2n} into even size blocks. - Geoffrey Critzer, Dec 03 2012
Except for a(0), row sums of A241171. - Peter Bala, Aug 20 2014
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255928. - Peter Bala, Mar 13 2015
Also the 2-packed words of degree n; cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words for 0<=k<=5. - Peter Luschny, Jul 06 2015

Seiichi Manyama, Table of n, a(n) for n = 0..235 (terms 0..40 from Peter Luschny)
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.

Cf. A241171, A210676, A210657, A028296, A210672, A210674, A255928.

f:=proc(n,k) option remember;  local i;
if n=0 then 1
else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
g:=k->[seq(f(n,k),n=0..40)];g(1); # N. J. A. Sloane, Mar 28 2012

nn=30;Select[Range[0,nn]!CoefficientList[Series[1/(2-Cosh[x]),{x,0,nn}],x],#>0&]  (* Geoffrey Critzer, Dec 03 2012 *)
a[0]=1; a[n_] := Sum[1/2*(1+(-1)^(2*n))*Sum[((-1)^(k-j)*Binomial[k, j]*Sum[(j-2*i )^(2*n)*Binomial[j, i], {i, 0, j}])/2^j, {j, 1, k}], {k, 1, n}]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)

a(n):=b(2*n+2);
b(n):=sum(((sum(((sum((j-2*i)^n*binomial(j,i),i,0,j))*(-1)^(k-j)*binomial(k,j))/2^(j),j,1,k))*((-1)^n+1))/2,k,1,n/2); /* Vladimir Kruchinin, Apr 23 2011 */

a(n):=sum(sum((i-k)^(2*n)*binomial(2*k,i)*(-1)^(i),i,0,k-1)/(2^(k-1)),k,1,2*n); /* Vladimir Kruchinin, Oct 05 2012 */

a(n) = if (n == 0, 1, sum(k=1, n, binomial(2*n, 2*n-2*k)*a(n-k)));

def A094088(n) :
    @CachedFunction
    def intern(n) :
        if n == 0 : return 1
        if n % 2 != 0 : return 0
        return add(intern(k)*binomial(n,k) for k in range(n)[::2])
    return intern(2*n)
[A094088(n) for n in (0..14)]  # Peter Luschny, Jul 14 2012

1/(2-cosh(x)) = Sum_{n>=0} a(n)x^(2n)/(2n)! = 1 + x^2/2 + 7x^4/24 + 121x^6/720 + ...
Recurrence: a(0)=1, a(n) = Sum_{k=1..n} C(2n, 2n-2k)*a(n-k).
a(0)=1 and, for n>=1, a(n)=b(2*n) where b(n) = sum(k=1..n/2,((sum(j=1..k, ((sum(i=0..j,(j-2*i)^n*binomial(j,i)))*(-1)^(k-j)*binomial(k,j))/2^(j)))*((-1)^n+1))/2). - Vladimir Kruchinin, Apr 23 2011
E.g.f.: 1/(2-cosh(x))=8*(1-x^2)/(8 - 12*x^2 + x^4*U(0))  where U(k)= 1 + 4*(k+1)*(k+2)/(2*k+3 - x^2*(2*k+3)/(x^2 + 8*(k+1)*(k+2)*(k+3)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 30 2012
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), n>0, a(0)=1. - Vladimir Kruchinin, Oct 05 2012
a(n) ~ 2*(2*n)! /(sqrt(3) * (log(2+sqrt(3)))^(2*n+1)). - Vaclav Kotesovec, Oct 19 2013

Corrected definition, Joerg Arndt, Apr 26 2011

A278074 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 4.

Original entry on oeis.org

1, 0, 1, 0, 1, 70, 0, 1, 990, 34650, 0, 1, 16510, 2702700, 63063000, 0, 1, 261630, 213519150, 17459442000, 305540235000, 0, 1, 4196350, 17651304000, 4350310965000, 231905038365000, 3246670537110000

Offset: 0

Peter Luschny, Jan 22 2017

			Triangle starts:
[1]
[0, 1]
[0, 1,     70]
[0, 1,    990,     34650]
[0, 1,  16510,   2702700,    63063000]
[0, 1, 261630, 213519150, 17459442000, 305540235000]

Robert Israel, Table of n, a(n) for n = 0..8510

Cf. A014608 (diagonal), A243665 (row sums), A211212 (alternating row sums), A281480 (central coefficients).
Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278073 (m=3).
Cf. A327024 (refinement).

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:
for n from 0 to 6 do PolynomialTools:-CoefficientList(P(4,n), x) od;
# Alternatively:
A278074_row := proc(n) 1/(1-t*((cosh(x)+cos(x))/2-1)); expand(series(%,x,4*n+1));
(4*n)!*coeff(%,x,4*n); PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 5 do A278074_row(n) od;

With[{m = 4}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 24, m}]];
Function[arg, CoefficientList[arg, t]] /@ % // Flatten

# uses [P from A278073]
def A278074_row(n): return list(P(4, n))
for n in (0..6): print(A278074_row(n)) # Peter Luschny, Mar 24 2020

E.g.f.: 1/(1-t*((cosh(x)+cos(x))/2-1)), nonzero terms.

A243664 Number of 3-packed words of degree n.

Original entry on oeis.org

1, 1, 21, 1849, 426405, 203374081, 173959321557, 242527666641289, 514557294036701349, 1577689559404884503761, 6714435826042791310638741, 38401291553086405072860452569, 287412720357301174793668207559205, 2753382861926383584939774967275568801

Offset: 0

N. J. A. Sloane, Jun 14 2014

nonn

See Novelli-Thibon (2014) for precise definition.

Andrew Howroyd, Table of n, a(n) for n = 0..100 (terms n = 0..30 from Peter Luschny)
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 16.

Cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words of degree n for 0<=k<=5.

g := t -> (exp(t)+2*exp(-t/2)*cos(sqrt(3)*t/2))/3: series(1/(2-g(t^(1/3))),t,14): seq(((3*n)!*coeff(%,t,n)),n=0..13); # Peter Luschny, Jul 07 2015

g[t_] := (Exp[t] + 2 Exp[-t/2] Cos[Sqrt[3] t/2])/3;
a[n_] := (3n)! SeriesCoefficient[1/(2 - g[t^(1/3)]), {t, 0, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 13 2018, after Peter Luschny *)

seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n]=sum(k=1, n, binomial(3*n, 3*k) * a[1+n-k])); a} \\ Andrew Howroyd, Jan 21 2020

def CEN(m, len):
    f, e, r, u = [1], [1], [1], 1
    for i in (1..len-1):
        f.append(rising_factorial(u, m))
        for k in range(i-1, -1, -1):
            e[k] = (e[k]*f[i])//f[i-k]
        s = sum(e); e.append(s); r.append(s)
        u += m
    return r
A243664 = lambda len: CEN(3,len)
A243664(14) # Peter Luschny, Jul 06 2015

# Alternative
def PackedWords3(n):
    shapes = [[x**3 for x in p] for p in Partitions(n)]
    return sum([factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes])
[PackedWords3(n) for n in (0..13)] # Peter Luschny, Aug 02 2015

a(n) = (3*n)! * [t^n] 1/(2-g(t^(1/3))) with g(t) = (exp(t)+2*exp(-t/2)*cos(sqrt(3)*t/2))/3. - Peter Luschny, Jul 07 2015

a(0) = 1; a(n) = Sum_{k=1..n} binomial(3*n,3*k) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020

a(0)=1 prepended, more terms from Peter Luschny, Jul 06 2015

A243666 Number of 5-packed words of degree n.

Original entry on oeis.org

1, 1, 253, 762763, 11872636325, 633287284180541, 90604069581412784683, 29529277377602939454694793, 19507327717978242212109900308085, 23927488379043876045061553841299192011, 50897056444296458534155179226333868898628813, 177758773838827813873239281786548960244155096117573

Offset: 0

N. J. A. Sloane, Jun 14 2014

nonn

See Novelli-Thibon (2014) for precise definition.

Andrew Howroyd, Table of n, a(n) for n = 0..100 (terms n = 0..30 from Peter Luschny)
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 16.

Cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words of degree n for 0<=k<=5.

a := (5+sqrt(5))/4: b := (5-sqrt(5))/4: g := t -> (exp(t)+2*exp(t-a*t)*cos(t*sqrt(b/2))+2*exp(t-b*t)*cos(t*sqrt(a/2)))/5: series(1/(2-g(t)),t,56): seq((5*n)!*(coeff(simplify(%),t,5*n)),n=0..11); # Peter Luschny, Jul 07 2015

b = (5 - Sqrt[5])/4; c = (5 + Sqrt[5])/4;
g[t_] := (Exp[t] + 2*Exp[t - c*t]*Cos[t*Sqrt[b/2]] + 2*Exp[t - b*t]* Cos[t*Sqrt[c/2]])/5;
a[n_] := (5n)! SeriesCoefficient[1/(2 - g[t]), { t, 0, 5 n}] // Simplify;
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 12}] (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n]=sum(k=1, n, binomial(5*n, 5*k) * a[1+n-k])); a} \\ Andrew Howroyd, Jan 21 2020

# uses[CEN from A243664]
A243666 = lambda len: CEN(5,len)
A243666(12) # Peter Luschny, Jul 06 2015

# Alternatively:
def PackedWords5(n):
    shapes = ([x*5 for x in p] for p in Partitions(n))
    return sum(factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
[PackedWords5(n) for n in (0..11)] # Peter Luschny, Aug 02 2015

a(0) = 1; a(n) = Sum_{k=1..n} binomial(5*n,5*k) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020

a(0)=1 prepended, more terms from Peter Luschny, Jul 06 2015

A352429 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n,4k+1) a(n-4*k-1).

Original entry on oeis.org

1, 1, 2, 6, 24, 121, 732, 5166, 41664, 378001, 3810512, 42253926, 511139904, 6698457481, 94535404992, 1429477706286, 23056267551744, 395120495014561, 7169579673404672, 137321623511274246, 2768602189953629184, 58609968225266985241, 1299827736206335767552, 30137364376923272989806

Offset: 0

Ilya Gutkovskiy, Mar 16 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000670, A006154, A243665, A291975, A306347, A352428, A352430.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 4 k + 1] a[n - 4 k - 1], {k, 0, Floor[(n - 1)/4]}]; Table[a[n], {n, 0, 23}]
nmax = 23; CoefficientList[Series[1/(1 - Sum[x^(4 k + 1)/(4 k + 1)!, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\4, x^(4*k+1)/(4*k+1)!)))) \\ Seiichi Manyama, Mar 23 2022

E.g.f.: 1 / (1 - Sum_{k>=0} x^(4*k+1) / (4*k+1)!).

E.g.f.: 1 / (1 - (sin(x) + sinh(x)) / 2).

A260883 Number of m-shape ordered 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, 3, 1, 1, 3, 9, 1, 1, 7, 13, 35, 1, 1, 21, 121, 75, 161, 1, 1, 71, 1849, 3907, 541, 913, 1, 1, 253, 35641, 426405, 202741, 4683, 6103, 1, 1, 925, 762763, 65782211, 203374081, 15430207, 47293, 47319, 1, 1, 3433, 17190265, 11872636325, 323213457781, 173959321557

Offset: 1

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. It is ordered if the positions of the blocks are taken into account.
If m = 0, all possible sizes are zero. Thus the number of ordered set partitions of 0-shape is the number of ordered partitions of n (partition numbers A101880).
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 ordered set partitions of 1-shape is a Fubini number (sequence A000670).
If m = 2, the set is {1, 2, ..., 2n} and the number of ordered set partitions of 2-shape is also the number of 2-packed words of degree n (sequence A094088).

			[ n ] [0  1   2      3         4            5                  6]
[ m ] -----------------------------------------------------------
[ 0 ] [1, 1,  3,     9,       35,          161,              913]  A101880
[ 1 ] [1, 1,  3,    13,       75,          541,             4683]  A000670
[ 2 ] [1, 1,  7,   121,     3907,       202741,         15430207]  A094088
[ 3 ] [1, 1, 21,  1849,   426405,    203374081,     173959321557]  A243664
[ 4 ] [1, 1, 71, 35641, 65782211, 323213457781, 3482943541940351]  A243665
        A244174
For example the number of ordered set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] is 1, 168 and 1680 respectively. Thus A(3,3) = 1849.
Formatted as a triangle:
[1]
[1, 1]
[1, 1, 3]
[1, 1, 3, 9]
[1, 1, 7, 13, 35]
[1, 1, 21, 121, 75, 161]
[1, 1, 71, 1849, 3907, 541, 913]
[1, 1, 253, 35641, 426405, 202741, 4683, 6103]

Without order: A260876.

Cf. A000670, A094088, A101880, A243664, A243665, A243666, A244174.

def A260883(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return sum(factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
for m in (0..4): print([A260883(m, n) for n in (0..6)])

From Petros Hadjicostas, Aug 02 2019: (Start)

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]) * multinomial(a_1 + ... + a_n, [a_1,..., 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)

A326585 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 4 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 36, 35, 0, 12046, 17820, 5775, 0, 16674906, 30263480, 16216200, 2627625, 0, 65544211366, 135417565890, 93516348900, 26189163000, 2546168625, 0, 588586227465426, 1334168329550300, 1083314031995250, 402794176785000, 69571511509500, 4509264634875

Offset: 0

Peter Luschny, Jul 21 2019

			Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 36, 35]
[3] [0, 12046, 17820, 5775]
[4] [0, 16674906, 30263480, 16216200, 2627625]
[5] [0, 65544211366, 135417565890, 93516348900, 26189163000, 2546168625]
[6] [0, 588586227465426, 1334168329550300, 1083314031995250, 402794176785000, 69571511509500, 4509264634875]

Row sums A243665. Main diagonal A025036.

A129062 (m=1, associated with A131689), A326477 (m=2, associated with A241171), A326587 (m=3, associated with A278073), this sequence (m=4, associated with A278074).

Maple
```
# See A326477.
```

T(n, k) = T_{4}(n, k) where T_{m}(n, k) is defined in A326477.

A327024 Ordered set partitions of the set {1, 2, ..., 4*n} with all block sizes divisible by 4, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

Original entry on oeis.org

1, 1, 1, 70, 1, 990, 34650, 1, 3640, 12870, 2702700, 63063000, 1, 9690, 251940, 26453700, 187065450, 17459442000, 305540235000, 1, 21252, 1470942, 2704156, 154448910, 8031343320, 9465511770, 374796021600, 3975514943400, 231905038365000, 3246670537110000

Offset: 0

Peter Luschny, Aug 27 2019

T_{m}(n, k) gives the number of ordered set partitions of the set {1, 2, ..., m*n} into sized blocks of shape m*P(n, k), where P(n, k) is the k-th integer partition of n in the 'canonical' order A080577. Here we assume the rows of A080577 to be 0-based and m*[a, b, c,..., h] = [m*a, m*b, m*c,..., m*h]. Here is case m = 4. For instance 4*P(4, .) = [[16], [12, 4], [8, 8], [8, 4, 4], [4, 4, 4, 4]].

			Triangle starts (note the subdivisions by ';' (A072233)):
[0] [1]
[1] [1]
[2] [1;    70]
[3] [1;   990;   34650]
[4] [1;  3640,   12870;  2702700;  63063000]
[5] [1;  9690,  251940; 26453700, 187065450; 17459442000; 305540235000]
[6] [1; 21252, 1470942,  2704156; 154448910,  8031343320,   9465511770;
     374796021600, 3975514943400; 231905038365000; 3246670537110000]
.
T(4, 1) = 3640 because [12, 4] is the integer partition 4*P(4, 1) in the canonical order and there are 1820 set partitions which have the shape [12, 4]. Finally, since the order of the sets is taken into account, one gets 2!*1820 = 3640.

Row sums: A243665, alternating row sums: A211212, main diagonal: A014608, central column: A281480, by length: A278074.
Cf. A178803 (m=0), A133314 (m=1), A327022 (m=2), A327023 (m=3), this sequence (m=4).
Cf. A080577, A000041, A072233.

# uses[GenOrdSetPart from A327022]
def A327024row(n): return GenOrdSetPart(4, n)
for n in (0..6): print(A327024row(n))

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A094088 E.g.f. 1/(2-cosh(x)) (even coefficients).

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A278074 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)* P(m, n-k)*z with P(m, 0) = 1 and m = 4.

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A243664 Number of 3-packed words of degree n.

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A243666 Number of 5-packed words of degree n.

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A352429 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n,4*k+1) * a(n-4*k-1).

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A260883 Number of m-shape ordered set partitions, square array read by ascending antidiagonals, A(m, n) for m, n >= 0.

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A326585 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 4 and 0 <= k <= n.

A278074 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 4.

A352429 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n,4k+1) a(n-4*k-1).