A000670 -id:A000670 - cp's OEIS frontend

A217391 Partial sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.

1, 2, 11, 180, 5805, 298486, 22228975, 2258856824, 300194704049, 50529463186170, 10505093602625139, 2643441560563225468, 791779611505017309493, 278371498870260182630654, 113516551713466910954246903, 53143864598655784249290736512, 28309328562668956145157858372537

Offset: 0

Emanuele Munarini, Oct 02 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000670, A006957, A005649, A217388, A217389, A217392.

Partial sums of A122725.

A000670:=func;
[&+[A000670(k)^2: k in [0..n]]: n in [0..14]]; // Bruno Berselli, Oct 03 2012

t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[t[k]^2, {k, 0, n}], {n, 0, 100}]

t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(t(k)^2,k,0,n),n,0,40);

for(n=0,30, print1(sum(k=0,n, (sum(j=0,k, j!*stirling(k,j,2)))^2), ", ")) \\ G. C. Greubel, Feb 07 2018

a(n) = Sum_{k=0..n} t(k)^2 where t = A000670 (ordered Bell numbers).

a(n) ~ (n!)^2 / (4 * (log(2))^(2*n+2)). - Vaclav Kotesovec, Nov 08 2014

A217392 Alternating sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.

Original entry on oeis.org

1, 0, 9, 160, 5465, 287216, 21643273, 2214984576, 295720862649, 49933547619472, 10404630591819497, 2622531836368780832, 786513638108085303193, 276793205620647080017968, 112961387008976003691598281, 52917386659933341334644891328, 28203267311410367019573922744697

Offset: 0

Emanuele Munarini, Oct 02 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000670, A006957, A005649, A217388, A217389, A217391.

A000670:=func;
[&+[(-1)^(n-k)*A000670(k)^2: k in [0..n]]: n in [0..14]]; // Bruno Berselli, Oct 03 2012

t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[(-1)^(n-k)t[k]^2, {k, 0, n}], {n, 0, 100}]

t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum((-1)^(n-k)*t(k)^2,k,0,n),n,0,40);

for(n=0,30, print1(sum(k=0,n, (-1)^(n-k)*(sum(j=0,k, j!*stirling(k,j,2)))^2), ", ")) \\ G. C. Greubel, Feb 07 2018

a(n) = sum((-1)^(n-k)*t(k)^2, k=0..n), where t = A000670 (ordered Bell numbers).

a(n) ~ (n!)^2 / (4 * (log(2))^(2*n+2)). - Vaclav Kotesovec, Nov 08 2014

A290352 Euler transform of the Fubini numbers (ordered Bell numbers, A000670).

Original entry on oeis.org

1, 1, 4, 17, 98, 678, 5687, 55656, 626161, 7963511, 113027113, 1770785023, 30346490633, 564546034917, 11327726548719, 243811768229012, 5602495216123312, 136878883607160468, 3542830077444873188, 96835203745704714722, 2787051847418347608600

Offset: 0

Alois P. Heinz, Jul 28 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..424

Cf. A000670, A007003, A095993, A290351.

b:= proc(n, m) option remember;
     `if`(n=0, m!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      b(d, 0), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);

b[n_, m_]:=b[n, m]=If[n==0, m!, Sum[b[n - 1, Max[m, j]], {j, m + 1}]]; a[n_]:=a[n]=If[n==0, 1, Sum[Sum[d*b[d, 0], {d, Divisors[j]}] a[n - j], {j, n}]/n]; Table[a[n], {n, 0, 50}] (* Indranil Ghosh, Jul 28 2017, after Maple code *)

a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, May 31 2019

A068942 a(n) = Bo(n^2), n=0,1..., where Bo(n) are the ordered Bell numbers, A000670.

Original entry on oeis.org

1, 1, 75, 7087261, 5315654681981355, 106697365438475775825583498141, 144199280951655469628360978109406917583513090155, 27656793065414932606012896651489726461435178241015434306518713649426461

Offset: 0

Karol A. Penson, Mar 09 2002

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..20

Cf. A000670, A068939, A249938, A249941.

a[n_] := PolyLog[-n^2, 1/2]/2; a[0] = 1; Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Mar 30 2016 *)
Table[Sum[k!*StirlingS2[n^2, k], {k, 0, n^2}], {n, 0, 10}] (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = sum(k=0, n^2, k!*stirling(n^2, k, 2)); \\ Seiichi Manyama, Jan 17 2022

a(n) = Sum_{k>=1} (k^(n^2))/2^(k+1); this is the analog of the Dobinski formula.
Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int(x^n*(sum(exp(-ln(x)^2/(4*ln(k))) / (2^k*sqrt(ln(k))), k=2..infinity)/(4*sqrt(Pi)*x)+Dirac(x-1)/4), x=0..infinity).
a(n) ~ (n^2)! / (2 * log(2)^(n^2 + 1)). - Vaclav Kotesovec, Jun 08 2021

A305852 Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).

Original entry on oeis.org

1, 1, 3, 16, 91, 658, 5567, 54917, 620081, 7905592, 112382245, 1762646331, 30231516786, 562750751610, 11297034281595, 243241826522376, 5591075279423398, 136633359995403580, 3537193288612096901, 96697587673174195740, 2783492094736121087958

Offset: 0

Alois P. Heinz, Jun 11 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..424

Cf. A000670, A290352, A305850, A305853.

g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(n, j), j=1..n))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

g[n_] := g[n] = If[n == 0, 1,
    Sum[g[n - j] Binomial[n, j], {j, 1, n}]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0,
    Sum[Binomial[g[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

G.f.: Product_{k>=1} (1+x^k)^A000670(k).

a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Sep 10 2019

A305853 Inverse Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).

Original entry on oeis.org

1, 3, 10, 62, 446, 3975, 41098, 484152, 6390488, 93419965, 1498268466, 26159940522, 494036061550, 10035451747919, 218207845446062, 5057251219752612, 124462048466812950, 3241773988594489244, 89093816361187396674, 2576652694087236419386, 78224564280680539732266

Offset: 1

Alois P. Heinz, Jun 11 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..424

Cf. A000670, A095993, A305846, A305852.

g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(n, j), j=1..n))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; g(n)-b(n, n-1) end:
seq(a(n), n=1..30);

g[n_] := g[n] = If[n == 0, 1,
    Sum[g[n - j] Binomial[n, j], {j, 1, n}]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
    Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = g[n] - b[n, n - 1];
a /@ Range[1, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} A000670(n) * x^n.

a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Sep 10 2019

A362585 Triangle read by rows, T(n, k) = A000670(n) * binomial(n, k).

Original entry on oeis.org

1, 1, 1, 3, 6, 3, 13, 39, 39, 13, 75, 300, 450, 300, 75, 541, 2705, 5410, 5410, 2705, 541, 4683, 28098, 70245, 93660, 70245, 28098, 4683, 47293, 331051, 993153, 1655255, 1655255, 993153, 331051, 47293, 545835, 4366680, 15283380, 30566760, 38208450, 30566760, 15283380, 4366680, 545835

Offset: 0

Peter Luschny, Apr 26 2023

			[0]    1;
[1]    1,     1;
[2]    3,     6,     3;
[3]   13,    39,    39,    13;
[4]   75,   300,   450,   300,    75;
[5]  541,  2705,  5410,  5410,  2705,   541;
[6] 4683, 28098, 70245, 93660, 70245, 28098, 4683;

Family of triangles: A055372 (m=0, Pascal), this sequence (m=1, Fubini), A362586 (m=2, Joffe), A362849 (m=3, A278073).

Cf. A000670 (column 0 and main diagonal), A216794 (row sums).

def TransOrdPart(m, n) -> list[int]:
    @cached_function
    def P(m: int, n: int):
        R = PolynomialRing(ZZ, "x")
        if n == 0: return R(1)
        return R(sum(binomial(m * n, m * k) * P(m, n - k) * x
                 for k in range(1, n + 1)))
    T = P(m, n)
    def C(k) -> int:
        return sum(T[j] * binomial(n, k) for j in range(n + 1))
    return [C(k) for k in range(n+1)]
def A362585(n) -> list[int]: return TransOrdPart(1, n)
for n in range(6): print(A362585(n))

A088791 Coefficient of x^n in A(x)^n is A000670(n), which gives preferential arrangements of n labeled elements.

Original entry on oeis.org

1, 1, 1, 2, 8, 46, 337, 2976, 30627, 359222, 4725968, 68903766, 1102712316, 19219507328, 362428546833, 7352854216056, 159705991698432, 3697928742242694, 90933523698184947, 2366758931071064064

Offset: 0

Paul D. Hanna, Oct 16 2003

Vaclav Kotesovec, Table of n, a(n) for n = 0..333

Cf. A000670, A084784.

nmax = 19; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^2 - (A[x A[x]] + x A[x]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=local(A, m); if(n<1, n==0, m=1; A=1+x; for(i=1, n, A=(subst(A, x, x*A+x*O(x^n)) + x*A)/A); polcoeff(A, n))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Feb 11 2015

G.f. satisfies: A(x)^2 = A(x*A(x)) + x*A(x).
a(n) ~ (n-1)! / (4 * (log(2))^(n+1)). - Vaclav Kotesovec, Feb 12 2015
O.g.f.: A(x) = x/( series reversion x*B(x) ), where B(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 137*x^5 + ... is the o.g.f. of A084784. - Peter Bala, Jun 23 2015

A122725 a(n) = A000670(n)^2.

Original entry on oeis.org

1, 1, 9, 169, 5625, 292681, 21930489, 2236627849, 297935847225, 50229268482121, 10454564139438969, 2632936466960600329, 789136169944454084025, 277579719258755165321161, 113238180214596650771616249, 53030348046942317338336489609, 28256184698070300360908567636025

Offset: 0

Vladeta Jovovic, Sep 23 2006

This is also the number of possible positions of n intervals on a line having a common non-punctual intersection. Proof: Let us denoted each interval Ai (1 <= i <= n) by the string AiAi. Then the set of all such relative positions is given by the S-language [A1 ⊗ A2 ... ⊗ An]^2. The cardinality of $A1 ⊗ A2 ... ⊗ An$ is given by A000670. - Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 26 2007

Cf. A000670, A055203, A101370.

b:= proc(n, k) option remember;
     `if`(n=0, 1, add(b(n-1, j)*j, j=k..k+1))
    end:
a:= n-> b(n, 0)^2:
seq(a(n), n=0..16);  # Alois P. Heinz, Aug 12 2025

Table[(PolyLog[ -z, 1/2]/2)^2, {z, 1, 16}] (* Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Oct 10 2006 *)

{a(n)=sum(k=0, n, stirling(n, k, 2)*k!)^2} \\ Paul D. Hanna, Nov 07 2009

a(n) = Sum_{m>=0} Sum_{k>=0} ((k*m)^n/2^(k+m+2)).
G.f.: Sum_{n>=0} (1/(2-exp(n*x))/2^(n+1)).
Sum_{n>=0} a(n)*log(1+x)^n/n! = o.g.f. of A101370. - Paul D. Hanna, Nov 07 2009
a(n) ~ (n!)^2 / (4 * (log(2))^(2*n+2)). - Vaclav Kotesovec, May 03 2015

More terms from Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Oct 10 2006

A163204 Triangle read by rows, A095989 convolved with A000670.

Original entry on oeis.org

1, 1, 2, 3, 2, 8, 13, 6, 8, 48, 75, 26, 24, 48, 368, 541, 150, 104, 144, 368, 3376, 4683, 1082, 600, 624, 1104, 3376, 35824, 47293, 9366, 4328, 3600, 4784, 10128, 35824, 430512, 545835, 94586, 37464, 25968, 27600, 43888, 107472, 430512, 5773936, 7087261, 1091670, 378344, 224784, 199088, 253200, 465712, 1291536, 5773936, 85482032

Offset: 1

Gary W. Adamson, Jul 23 2009

Row sums = A000670 starting with offset 1: (1, 3, 13, 75, 541, 4683,...).
Left border = A000670, right border = A095989.
Second column: A076726. - Michel Marcus, Mar 31 2016

			First few rows of the triangle are:
1;
1, 2;
3, 2, 8;
13, 6, 8, 48;
75, 26, 24, 48, 368;
541, 150, 104, 144, 368, 3376;
4683, 1082, 600, 624, 1104, 3376, 35824;
47293, 9366, 4328, 3600, 4784, 10128, 35824, 430512;
...

G. C. Greubel, Table of n, a(n) for n = 1..1225

Cf. A000670, A076726, A095989.

max = 10; Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k - i - r)!), {i, 0, k - r}], {k, r, n}]; Fubini[0, 1] = 1; A000670 = Table[ Fubini[n, 1], {n, 0, max}]; s = 1 - 1/Sum[Fubini[k, 1] q^k, {k, 0, max}] + O[q]^max; A095989 = CoefficientList[s/q, q]; row[n_] := A095989[[1 ;; n]]*Reverse[A000670[[1 ;; n]]]; Table[row[n], {n, 1, max-1}] // Flatten (* Jean-François Alcover, Mar 31 2016 *)

Descending antidiagonals of a multiplication table formed by convolving A095989 with A000670, where A095989 is the INVERTi transform of A000670 starting (1, 3, 13, 75,...).

a(23) corrected by Jean-François Alcover, Mar 31 2016

Terms a(37) onward added by G. C. Greubel, Dec 10 2016

cp's OEIS Frontend

A217391 Partial sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.

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A217392 Alternating sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.

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A290352 Euler transform of the Fubini numbers (ordered Bell numbers, A000670).

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A068942 a(n) = Bo(n^2), n=0,1..., where Bo(n) are the ordered Bell numbers, A000670.

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A305852 Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).

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A305853 Inverse Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).

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A362585 Triangle read by rows, T(n, k) = A000670(n) * binomial(n, k).

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A088791 Coefficient of x^n in A(x)^n is A000670(n), which gives preferential arrangements of n labeled elements.

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