A053492 -id:A053492 - cp's OEIS frontend

A317885 Number of series-reduced free pure achiral multifunctions with one atom and n positions.

1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 9, 14, 21, 32, 45, 69, 103, 153, 224, 338, 500, 746, 1107, 1645, 2447, 3652, 5413, 8052, 11993, 17834, 26500, 39447, 58655, 87240, 129772, 193001, 287034, 427014, 635048, 944501, 1404910, 2089633, 3107864, 4622670, 6875533

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A series-reduced free pure achiral multifunction (SRAM) is either (case 1) the leaf symbol "o", or (case 2) a nonempty and non-unitary expression of the form h[g, ..., g] where h and g are SRAMs. The number of positions in a SRAM is the number of brackets [...] plus the number of o's.

			The a(10) = 7 SRAMs:
  o[o[o,o],o[o,o]]
  o[o,o][o,o][o,o]
  o[o,o][o,o,o,o,o]
  o[o,o,o][o,o,o,o]
  o[o,o,o,o][o,o,o]
  o[o,o,o,o,o][o,o]
  o[o,o,o,o,o,o,o,o]

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A317853, A317875.

Cf. A317882, A317883, A317884.

a[n_]:=If[n==1,1,Sum[a[k]*Sum[a[d],{d,Most[Divisors[n-k-1]]}],{k,n-2}]];
Array[a,12]

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*sum(k=2, n-2, subst(p + O(x^(n\k+1)), x, x^k)) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, if(dAndrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1), d < n - k - 1} a(d).

Terms a(17) and beyond from Andrew Howroyd, Aug 19 2018

A259065 E.g.f.: Series_Reversion( 5x - 4x*exp(x) ).

Original entry on oeis.org

1, 8, 204, 8656, 514100, 39254904, 3663341724, 404021632928, 51413304278916, 7414832746025800, 1195170934203398636, 212923233266007511152, 41545466670049713766356, 8811212141812890158250776, 2018230889016461893216938300, 496523506149784085749952075584, 130578628540561635331879674437156

Offset: 1

Paul D. Hanna, Jun 17 2015

nonn

			E.g.f.: A(x) = x + 8*x^2/2! + 204*x^3/3! + 8656*x^4/4! + 514100*x^5/5! +...
where A(5*x - 4*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 8*x^2 + 204*x^3 + 8656*x^4 + 514100*x^5 + 39254904*x^6 +...
where F(x)/x = 1/5 + 4/(5-x)^2 + 4^2/(5-2*x)^3 + 4^3/(5-3*x)^4 + 4^4/(5-4*x)^5 +...

G. C. Greubel, Table of n, a(n) for n = 1..305

Cf. A053492, A259063, A259064, A259066.

Rest[CoefficientList[InverseSeries[Series[5*x - 4*x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)

{a(n) = local(A=x); A = serreverse(5*x - 4*x*exp(x +x*O(x^n) )); n!*polcoeff(A,n)}
for(n=1,20,print1(a(n),", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A = x+sum(m=1, n, Dx(m-1, 4^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A = x*exp(sum(m=1, n, Dx(m-1, 4^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

O.g.f.: x * Sum_{n>=0}  4^n / (5 - n*x)^(n+1).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 4^n * (exp(x)-1)^n * x^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 4^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) ~ (c/(5*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(5*exp(1)/4). - Vaclav Kotesovec, Jun 19 2015

A259066 E.g.f.: Series_Reversion( 6x - 5x*exp(x) ).

Original entry on oeis.org

1, 10, 315, 16520, 1212775, 114465780, 13204213435, 1800094703440, 283154358503295, 50478562633826300, 10057594831485171355, 2214859039031666012760, 534202513174577053611415, 140048168049127802257998820, 39652657811418543065286846075, 12058716801545122639605896216480

Offset: 1

Paul D. Hanna, Jun 17 2015

nonn

			E.g.f.: A(x) = x + 10*x^2/2! + 315*x^3/3! + 16520*x^4/4! + 1212775*x^5/5! +...
where A(6*x - 5*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 10*x^2 + 315*x^3 + 16520*x^4 + 1212775*x^5 +...
where F(x)/x = 1/6 + 5/(6-x)^2 + 5^2/(6-2*x)^3 + 5^3/(6-3*x)^4 + 5^4/(6-4*x)^5 +...

G. C. Greubel, Table of n, a(n) for n = 1..295

Cf. A053492, A259063, A259064, A259065.

Rest[CoefficientList[InverseSeries[Series[6*x - 5*x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)

{a(n) = local(A=x); A = serreverse(6*x - 5*x*exp(x +x*O(x^n) )); n!*polcoeff(A,n)}
for(n=1,20,print1(a(n),", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A = x + sum(m=1, n, Dx(m-1, 5^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A = x*exp(sum(m=1, n, Dx(m-1, 5^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

O.g.f.: x * Sum_{n>=0}  5^n / (6 - n*x)^(n+1).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 5^n * (exp(x)-1)^n * x^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 5^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) ~ (c/(6*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(6*exp(1)/5). - Vaclav Kotesovec, Jun 19 2015

A317853 a(1) = 1; a(n > 1) = Sum_{0 < k < n} (-1)^(n - k - 1) a(n - k) Sum_{d|k} a(d).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 6, 11, 14, 23, 26, 51, 70, 114, 147, 237, 314, 516, 715, 1118, 1549, 2353, 3252, 5011, 7235, 10724, 15142, 22504, 32506, 47770, 69173, 100980, 146657, 212504, 308563, 448256, 658037, 946166, 1373739, 1988283, 2919185, 4197886, 6118850

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317875.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.
Cf. A317882, A317883, A317884, A317885.

a[n_]:=a[n]=If[n==1,1,Sum[(-1)^(n-k-1)*a[n-k]*Sum[a[d],{d,Divisors[k]}],{k,n-1}]];
Array[a,50]

A305919 a(n) = n! * [x^n] 1/(2 - exp(x))^n.

Original entry on oeis.org

1, 1, 8, 99, 1704, 37625, 1014348, 32300359, 1186399952, 49376357109, 2296400723220, 118031059900523, 6643848377509368, 406471060412884753, 26856124898028246044, 1905791887135240982415, 144563460111417997403040, 11673024609379676114380877, 999663240630210837032231460

Offset: 0

Ilya Gutkovskiy, Jun 14 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..350
N. J. A. Sloane, Transforms

Cf. A000670, A005649, A053492, A226513, A226515.

Table[n! SeriesCoefficient[1/(2 - Exp[x])^n, {x, 0, n}], {n, 0, 18}]
Table[SeriesCoefficient[Sum[Binomial[n + k - 1, k] k! x^k/Product[1 - j x, {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Table[Sum[StirlingS2[n, k] Binomial[n + k - 1, k] k!, {k, 0, n}], {n, 0, 18}]

a(n) = [x^n] Sum_{k>=0} binomial(n+k-1,k)*k!*x^k/Product_{j=1..k} (1 - j*x).
a(n) = Sum_{k=0..n} Stirling2(n,k)*binomial(n+k-1,k)*k!.
a(n) ~ n! * c * ((1 + r)*(1 + 2*r))^n / sqrt(n), where r = (-1 + 1/(-1 + LambertW(2*exp(1))))/2 = 0.833964643008471735434624869020826957702396269585... is the root of the equation (2 + 1/r) * (1 + r*LambertW(-exp(-1/r)/r)) = 1 and c = 1/sqrt(2*Pi*(1 + LambertW(2*exp(1)))) = 0.258877607195571655640738032164006... Equivalently, a(n) ~ LambertW(2*exp(1))^n * n^n / (sqrt(1 + LambertW(2*exp(1))) * 2^n * exp(n) * (LambertW(2*exp(1)) - 1)^(2*n)). - Vaclav Kotesovec, Dec 15 2019, updated Mar 17 2024

A317676 Triangle whose n-th row lists in order all e-numbers of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 16, 7, 10, 12, 13, 21, 25, 27, 32, 36, 64, 81, 128, 256, 11, 14, 17, 18, 28, 33, 35, 41, 45, 49, 75, 93, 100, 125, 144, 145, 169, 216, 243, 279, 441, 512, 625, 729, 1024, 1296, 2048, 2187, 4096, 6561, 8192, 16384, 65536, 524288, 8388608, 9007199254740992

Offset: 1

Gus Wiseman, Aug 03 2018

Given a positive integer n we construct a unique free pure symmetric multifunction e(n) by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)].

Every free pure symmetric multifunction (with empty expressions allowed) f with one atom and n positions has a unique e-number n such that e(n) = f, and vice versa, so this sequence is a permutation of the positive integers.

			Triangle begins:
  1
  2
  3   4
  5   6   8   9  16
  7  10  12  13  21  25  27  32  36  64  81 128 256
Corresponding triangle of free pure symmetric multifunctions (with empty expressions allowed) begins:
  o,
  o[],
  o[][], o[o],
  o[][][], o[o][], o[o[]], o[][o], o[o,o].

Mathematica Reference, Orderless

Row lengths are A277996.

Cf. A007916, A052409, A052410, A052893, A053492, A215366, A279944, A280000, A299759, A317658, A317659, A317677.

maxUsing[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h],Union[Sort/@Tuples[maxUsing/@p]]}],{p,IntegerPartitions[g]}]]];
radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]==1];
Clear[rad];rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
ungo[x_?AtomQ]:=1;ungo[h_[g___]]:=rad[ungo[h]]^(Times@@Prime/@ungo/@{g});
Table[Sort[ungo/@maxUsing[n]],{n,5}]

A259062 E.g.f.: Series_Reversion( -x + 2xexp(-x) ).

Original entry on oeis.org

1, 4, 42, 728, 17630, 548532, 20852370, 936655792, 48540537702, 2850727359500, 187107038833946, 13572973331551944, 1078343465147156910, 93119965280416893028, 8684514946963752624930, 869915871265946242868576, 93146889134541855185069942, 10617155946603647157142073916

Offset: 1

Paul D. Hanna, Jun 18 2015

nonn

			E.g.f.: A(x) = x + 4*x^2/2! + 42*x^3/3! + 728*x^4/4! + 17630*x^5/5! +...
where A(-x + 2*x*exp(-x)) = x.

Cf. A258872, A053492, A259063, A259064, A259065, A259066.

Rest[CoefficientList[InverseSeries[Series[-x + 2*x*E^(-x), {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)

{a(n) = local(A=x); A = serreverse(-x + 2*x*exp(-x +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x*O(x^n)))^m*x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x*O(x^n)))^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^(n-1) / n! ).
a(n) ~ (1-c) * n^(n-1) / (sqrt(1+c) * (c + 1/c - 2)^n * exp(n)), where c = LambertW(exp(1)/2) = 0.685076942154593946... . - Vaclav Kotesovec, Jun 19 2015

A304485 Regular triangle where T(n,k) is the number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 4, 0, 1, 12, 23, 7, 0, 1, 20, 81, 73, 12, 0, 1, 30, 209, 407, 206, 19, 0, 1, 42, 451, 1566, 1751, 534, 30, 0, 1, 56, 858, 4711, 9593, 6695, 1299, 45, 0, 1, 72, 1494, 11951, 39255, 51111, 23530, 3004, 67, 0, 1, 90, 2430, 26752, 130220, 278570, 245319, 77205, 6664, 97, 0

Offset: 1

Gus Wiseman, Aug 17 2018

A free pure symmetric multifunction (with empty expressions allowed) f in EOME is either (case 1) a positive integer, or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where k >= 0, h is in EOME, each of the g_i for i = 1, ..., k is in EOME, and for i < j we have g_i <= g_j under a canonical total ordering of EOME, such as the Mathematica ordering of expressions.

T(n,k) is also the number of inequivalent colorings of orderless Mathematica expressions with n positions and k leaves.

			Inequivalent representatives of the T(5,3) = 23 Mathematica expressions:
  1[][1,1]  1[1,1][]  1[1][1]  1[1[1]]  1[1,1[]]
  1[][1,2]  1[1,2][]  1[1][2]  1[1[2]]  1[1,2[]]
  1[][2,2]  1[2,2][]  1[2][1]  1[2[1]]  1[2,1[]]
  1[][2,3]  1[2,3][]  1[2][2]  1[2[2]]  1[2,2[]]
                      1[2][3]  1[2[3]]  1[2,3[]]
Triangle begins:
    1
    1    0
    1    2    0
    1    6    4    0
    1   12   23    7    0
    1   20   81   73   12    0
    1   30  209  407  206   19    0
    1   42  451 1566 1751  534   30    0

Andrew Howroyd, Table of n, a(n) for n = 1..325 (rows 1..25)

Row sums are A300626.

Cf. A000612, A007716, A052893, A053492, A277996, A280000, A317676.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + x*p*sExp(p)); p}
T(n)={my(v=Vec(InequivalentColoringsSeq(sFuncSubst(cycleIndexSeries(n), i->sv(i)*y^i)))); vector(n, n, Vecrev(v[n]/y, n))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 01 2021

Terms a(37) and beyond from Andrew Howroyd, Jan 01 2021

A317659 Regular triangle where T(n,k) is the number of distinct free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 17, 7, 1, 0, 1, 15, 43, 33, 9, 1, 0, 1, 21, 92, 118, 55, 11, 1, 0, 1, 28, 174, 341, 252, 82, 13, 1, 0, 1, 36, 302, 845, 935, 463, 115, 15, 1, 0, 1, 45, 490, 1864, 2921, 2103, 769, 153, 17, 1, 0, 1, 55, 755

Offset: 1

Gus Wiseman, Aug 03 2018

			The T(5,3) = 5 expressions are o[o[o]], o[o,o[]], o[][o,o], o[o][o], o[o,o][].
Triangle begins:
    1
    1    0
    1    1    0
    1    3    1    0
    1    6    5    1    0
    1   10   17    7    1    0
    1   15   43   33    9    1    0
    1   21   92  118   55   11    1    0
    1   28  174  341  252   82   13    1    0
    1   36  302  845  935  463  115   15    1    0
    1   45  490 1864 2921 2103  769  153   17    1    0
    1   55  755 3755 7981 8012 4145 1187  197   19    1    0

Mathematica Reference, Orderless

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317654, A317655, A317656, A317676.

maxUsing[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h],Union[Sort/@Tuples[maxUsing/@p]]}],{p,IntegerPartitions[g]}]]];
Table[Length[Select[maxUsing[n],Length[Position[#,"o"]]==k&]],{n,12},{k,n}]

A368584 Table read by rows: T(n, k) = A124320(n + 1, k) * A048993(n, k).

Original entry on oeis.org

1, 0, 2, 0, 3, 12, 0, 4, 60, 120, 0, 5, 210, 1260, 1680, 0, 6, 630, 8400, 30240, 30240, 0, 7, 1736, 45360, 327600, 831600, 665280, 0, 8, 4536, 216720, 2772000, 13305600, 25945920, 17297280, 0, 9, 11430, 956340, 20207880, 162162000, 575134560, 908107200, 518918400

Offset: 0

Peter Luschny, Jan 10 2024

			Triangle starts:
  [0] [1]
  [1] [0, 2]
  [2] [0, 3,   12]
  [3] [0, 4,   60,    120]
  [4] [0, 5,  210,   1260,    1680]
  [5] [0, 6,  630,   8400,   30240,    30240]
  [6] [0, 7, 1736,  45360,  327600,   831600,   665280]
  [7] [0, 8, 4536, 216720, 2772000, 13305600, 25945920, 17297280]

Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Cf. A124320 (rising factorial), A048993(Stirling2), A053492 (row sums), A213236 (alternating row sums), A001813 (main diagonal), A368583.

def Trow(n): return [rising_factorial(n+1, k)*stirling_number2(n, k) for k in range(n+1)]
for n in range(7): print(Trow(n))

cp's OEIS Frontend

A317885 Number of series-reduced free pure achiral multifunctions with one atom and n positions.

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A259065 E.g.f.: Series_Reversion( 5*x - 4*x*exp(x) ).

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A259066 E.g.f.: Series_Reversion( 6*x - 5*x*exp(x) ).

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A317853 a(1) = 1; a(n > 1) = Sum_{0 < k < n} (-1)^(n - k - 1) a(n - k) Sum_{d|k} a(d).

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A305919 a(n) = n! * [x^n] 1/(2 - exp(x))^n.

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A317676 Triangle whose n-th row lists in order all e-numbers of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

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A259062 E.g.f.: Series_Reversion( -x + 2*x*exp(-x) ).

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Mathematica

PARI

PARI

PARI

Formula

A304485 Regular triangle where T(n,k) is the number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions and k leaves.

A259065 E.g.f.: Series_Reversion( 5x - 4x*exp(x) ).

A259066 E.g.f.: Series_Reversion( 6x - 5x*exp(x) ).

A259062 E.g.f.: Series_Reversion( -x + 2xexp(-x) ).