A059606 -id:A059606 - cp's OEIS frontend

A035098 Near-Bell numbers: partitions of an n-multiset with multiplicities 1, 1, 1, ..., 1, 2.

1, 2, 4, 11, 36, 135, 566, 2610, 13082, 70631, 407846, 2504071, 16268302, 111378678, 800751152, 6027000007, 47363985248, 387710909055, 3298841940510, 29119488623294, 266213358298590, 2516654856419723, 24566795704844210

Offset: 1

George Beck

nonn

A035098 and A000070 are near the two ends of a spectrum. Another way to look at A000070 is as the number of partitions of an n-multiset with multiplicities n-1, 1.
The very ends are the number of partitions and the Stirling numbers of the second kind, which count the n-multiset partitions with multiplicities n and 1,1,1,...,1, respectively.
Intermediate sequences are the number of ways of partitioning an n-multiset with multiplicities some partition of n.

			a(3)=4 because there are 4 ways to partition the multiset {1,2,2} (with multiplicities {1,2}): {{1,2,2}} {{1,2},{2}} {{1},{2,2}} {{1},{2},{2}}.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018

Alois P. Heinz, Table of n, a(n) for n = 1..576 (first 200 terms from Vincenzo Librandi)
M. Griffiths, Generalized Near-Bell Numbers, JIS 12 (2009) 09.5.7.
M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
Martin Griffiths, Generating Functions for Extended Stirling Numbers of the First Kind, Journal of Integer Sequences, 17 (2014), #14.6.4.

Cf. A000070, A000110, A059606, A087648.
Row sums of A241500.
Column 1 of array in A322765.
Row n=2 of A346426.

with(combinat): a:= n-> floor(1/2*(bell(n-2)+bell(n-1)+bell(n))): seq(a(n), n=1..25); # Zerinvary Lajos, Oct 07 2007

f[n_] := Sum[ StirlingS2[n, k] ((k + 1) (k + 2)/2 + 1), {k, 0, n}]; Array[f, 22, 0]
f[n_] := (BellB[n] + BellB[n + 1] + BellB[n + 2])/2; Array[f, 22, 0]
Range[0, 21]! CoefficientList[ Series[ (1 + Exp@ x)^2/2 Exp[ Exp@ x - 1], {x, 0, 21}], x] (* 3 variants by Robert G. Wilson v, Jan 13 2011 *)
Join[{1},Total[#]/2&/@Partition[BellB[Range[0,30]],3,1]] (* Harvey P. Dale, Jan 02 2019 *)

Sum_{k=0..n} Stirling2(n, k)*((k+1)*(k+2)/2+1). E.g.f.: 1/2*(1+exp(x))^2*exp(exp(x)-1). (1/2)*(Bell(n)+Bell(n+1)+Bell(n+2)). - Vladeta Jovovic, Sep 23 2003 [for offset -1]

a(n) ~ Bell(n)/2 * (1 + LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

More terms from Vladeta Jovovic, Sep 23 2003

A087648 a(n) = (1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)).

Original entry on oeis.org

1, 3, 9, 31, 120, 514, 2407, 12205, 66491, 386699, 2388096, 15589732, 107165081, 773106715, 5836100685, 45981026703, 377230766908, 3215977070706, 28437411817135, 260380616093533, 2464930698184351, 24091925888687459, 242802079705721156, 2520198597834860148

Offset: 0

Vladeta Jovovic, Sep 23 2003

nonn

Sum of last number in all set partitions of n+1. E.g. The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}, so a(2) = 1+2+1+2+3 = 9. - Franklin T. Adams-Watters, Jun 07 2006

Number of partitions of the (n+2)-multiset {0,0,1,2,...,n} into distinct multisets. Also number of factorizations of 2 * Product_{i=1..n+1} prime(i) into distinct factors. - Alois P. Heinz, Jul 30 2021

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

Cf. A000110, A035098, A059606.
Main diagonal of A120057, row sums of A120095.
Column 1 of array in A322770.
Row n=2 of A346520.

[(1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)) : n in [0..30]]; // Vincenzo Librandi, Nov 13 2011

f[0]=1; f[n_] := Sum[ StirlingS2[n, k]*Binomial[k+2, k ], {k, 1, n}]; Table[ f[n], {n, 0, 20}] (* Zerinvary Lajos, Mar 31 2007 *)
(#[[3]]+#[[2]]-#[[1]])/2&/@Partition[BellB[Range[0,30]],3,1] (* Harvey P. Dale, Jul 20 2021 *)

A059604 Coefficients of polynomials (n-1)!P(n,k), P(n,k) = Sum_{i=0..n} Stirling2(n,i)binomial(k+i-1,k).

Original entry on oeis.org

1, 1, 2, 1, 9, 10, 1, 24, 107, 90, 1, 50, 575, 1750, 1248, 1, 90, 2135, 16050, 38244, 24360, 1, 147, 6265, 95445, 537334, 1078728, 631440, 1, 224, 15610, 424340, 4734289, 21569996, 38105220, 20865600, 1, 324, 34482, 1529640, 30128049

Offset: 1

Vladeta Jovovic, Jan 29 2001

			[1],
[1, 2],
[1, 9, 10],
[1, 24, 107, 90],
[1, 50, 575, 1750, 1248],
[1, 90, 2135, 16050, 38244, 24360],
[1, 147, 6265, 95445, 537334, 1078728, 631440],
...
P(2,k) = k + 2,
P(3,k) = (1/2!)*(k^2 + 9*k + 10),
P(4,k) = (1/3!)*(k^3 + 24*k^2 + 107*k + 90).

Vladeta Jovovic, More information

Cf. A059605, A059606, A320962.

P := (n, k) -> (n-1)!*add(Stirling2(n,i)*binomial(k+i-1,k), i=0..n):
for n from 1 to 8 do seq(coeff(expand(P(n,x)),x,n-k), k=1..n) od; # Peter Luschny, Nov 07 2018

row[n_] := (n-1)! CoefficientList[Sum[StirlingS2[n,i] Binomial[k+i-1,k] // FunctionExpand, {i,0,n}], k] // Reverse;
Array[row,10] // Flatten (* Jean-François Alcover, Jun 03 2019 *)

row(n)={Vec((n-1)!*sum(i=0, n, stirling(n,i,2)*binomial(x+i-1,i-1)))}
for(n=1, 10, print(row(n))) \\ Andrew Howroyd, Nov 07 2018

A059605 a(n) = (1/3!)(n^3 + 24n^2 + 107*n + 90), compare A059604.

Original entry on oeis.org

15, 37, 68, 109, 161, 225, 302, 393, 499, 621, 760, 917, 1093, 1289, 1506, 1745, 2007, 2293, 2604, 2941, 3305, 3697, 4118, 4569, 5051, 5565, 6112, 6693, 7309, 7961, 8650, 9377, 10143, 10949, 11796, 12685, 13617, 14593, 15614, 16681, 17795, 18957

Offset: 0

Vladeta Jovovic, Jan 29 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
More information

Cf. A052905, A059340, A059604, A059606.

[(1/6)*(n^3+24*n^2+107*n+90) : n in [0..50]]; // Vincenzo Librandi, Nov 13 2011

G.f.: (15 - 23*x + 10*x^2 - x^3)/(1-x)^4, compare A059340.

More terms from James Sellers, Feb 01 2001

A087649 a(n) = (1/2)*(Bell(n+2)-Bell(n+1)+Bell(n)).

Original entry on oeis.org

1, 2, 6, 21, 83, 363, 1733, 8942, 49484, 291871, 1825501, 12054705, 83734241, 609851830, 4644041462, 36883843101, 304846039251, 2616765134351, 23286746418237, 214489200063218, 2041785040262972, 20060079966396887, 203156789589084133, 2118391734395139205

Offset: 0

Vladeta Jovovic, Sep 23 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.

Cf. A000110, A035098, A059606.

[(1/2)*(Bell(n+2)-Bell(n+1)+Bell(n)): n in [0..30]]; // Vincenzo Librandi, Nov 13 2011

G.f.: (1-x+x^2)/(2*x*Q(0)) - 1/(2*x) + 1/2, where Q(k)= 1 - x - x/(1 - x*(2*k+1)/(1 - x - x/(1 - x*(2*k+2)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, May 13 2013

E.g.f.: exp(exp(x) - 1) * (exp(2*x) + 1) / 2. - Ilya Gutkovskiy, Aug 09 2021

A323618 Expansion of e.g.f. (1 + x)log(1 + x)(2 + log(1 + x))/2.

Original entry on oeis.org

0, 1, 2, -1, 1, -1, -2, 34, -324, 2988, -28944, 300816, -3371040, 40710240, -528439680, 7348717440, -109109064960, 1723814265600, -28888702617600, 512030734387200, -9572240647065600, 188274945999974400, -3887144020408320000, 84062926436751360000, -1900475323780239360000

Offset: 0

Ilya Gutkovskiy, Jan 20 2019

sign

Cf. A000217, A045406, A048994, A059606, A081052.

[(&+[StirlingFirst(n,k)*Binomial(k+1,2): k in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 07 2019

f:= gfun:-rectoproc({a(n) =  (5-2*n)*a(n-1) - (n-3)^2*a(n-2), a(0)=0, a(1)=1, a(2)=2, a(3)=-1}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jan 20 2019

nmax = 24; CoefficientList[Series[(1 + x) Log[1 + x] (2 + Log[1 + x])/2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k (k + 1)/2, {k, 0, n}], {n, 0, 24}]
Join[{0,1,2,-1}, RecurrenceTable[{a[n]==(5-2*n)*a[n-1]-(n-3)^2*a[n-2], a[2]==2, a[3]==-1}, a, {n,4,25}]] (* G. C. Greubel, Feb 07 2019 *)

{a(n) = sum(k=0,n, stirling(n,k,1)*binomial(k+1,2))};
vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 07 2019

[sum((-1)^(k+n)*stirling_number1(n,k)*binomial(k+1,2) for k in (0..n)) for n in (0..25)] # G. C. Greubel, Feb 07 2019

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000217(k).
a(n) ~ -(-1)^n * log(n) * n! / n^2 * (1 + (gamma - 2)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 20 2019
a(n) =  (5-2*n)*a(n-1) - (n-3)^2*a(n-2) for n >= 4. - Robert Israel, Jan 20 2019

A372624 Expansion of e.g.f. exp(1 - exp(x)) * (exp(x) - 1)^2 / 2.

Original entry on oeis.org

0, 0, 1, 0, -5, -10, 16, 154, 365, -750, -9749, -35222, 20956, 1013220, 6007821, 10272092, -129948837, -1405396426, -6318145964, 7407235766, 371429230721, 3172609248526, 11070816858267, -73488239926510, -1500342260080360, -11917913896465720, -31231507292803479

Offset: 0

Ilya Gutkovskiy, May 07 2024

sign

Cf. A000217, A000225, A000587, A003128, A059606, A081047, A101851.

nmax = 26; CoefficientList[Series[Exp[1 - Exp[x]] (Exp[x] - 1)^2/2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n, k] Binomial[k, 2], {k, 0, n}], {n, 0, 26}]

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * binomial(k,2).

a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,2) * A000587(n-k).

cp's OEIS Frontend

A035098 Near-Bell numbers: partitions of an n-multiset with multiplicities 1, 1, 1, ..., 1, 2.

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A087648 a(n) = (1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)).

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Magma

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A059604 Coefficients of polynomials (n-1)!*P(n,k), P(n,k) = Sum_{i=0..n} Stirling2(n,i)*binomial(k+i-1,k).

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A059605 a(n) = (1/3!)*(n^3 + 24*n^2 + 107*n + 90), compare A059604.

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Magma

Formula

Extensions

A087649 a(n) = (1/2)*(Bell(n+2)-Bell(n+1)+Bell(n)).

Views

Author

Keywords

Links

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Programs

Magma

Formula

A323618 Expansion of e.g.f. (1 + x)*log(1 + x)*(2 + log(1 + x))/2.

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Keywords

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A372624 Expansion of e.g.f. exp(1 - exp(x)) * (exp(x) - 1)^2 / 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A059604 Coefficients of polynomials (n-1)!P(n,k), P(n,k) = Sum_{i=0..n} Stirling2(n,i)binomial(k+i-1,k).

A059605 a(n) = (1/3!)(n^3 + 24n^2 + 107*n + 90), compare A059604.

A323618 Expansion of e.g.f. (1 + x)log(1 + x)(2 + log(1 + x))/2.