A271703
Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.
Original entry on oeis.org
1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 36, 12, 1, 0, 120, 240, 120, 20, 1, 0, 720, 1800, 1200, 300, 30, 1, 0, 5040, 15120, 12600, 4200, 630, 42, 1, 0, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, 0, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1
Offset: 0
As a rectangular array (diagonals of the triangle):
1, 1, 1, 1, 1, 1, ... A000012
0, 2, 6, 12, 20, 30, ... A002378
0, 6, 36, 120, 300, 630, ... A083374
0, 24, 240, 1200, 4200, 11760, ... A253285
0, 120, 1800, 12600, 58800, 211680, ...
0, 720, 15120, 141120, 846720, 3810240, ...
A000007, A000142, A001286, A001754, A001755, A001777.
The triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 0 1
2: 0 2 1
3: 0 6 6 1
4: 0 24 36 12 1
5: 0 120 240 120 20 1
6: 0 720 1800 1200 300 30 1
7: 0 5040 15120 12600 4200 630 42 1
8: 0 40320 141120 141120 58800 11760 1176 56 1
9: 0 362880 1451520 1693440 846720 211680 28224 2016 72 1
10: 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1
... - _Wolfdieter Lang_, Jun 12 2017
- R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., pp. 312, 552.
- I. Lah, Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik, Mitt.-Bl. Math. Statistik, 7:203-213, 1955.
- T. Mansour, M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, CRC Press, 2016
- Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
- Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
- M. F. Hasler and Peter Luschny, Formulas for A271703, OEIS Wiki, Aug. 2017.
- S. A. Joni, G.-C. Rota, and B. Sagan, From sets to functions: Three elementary examples, Discrete Mathematics, Volume 37, Issues 2-3, 1981, 193-202.
- Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
- D. E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78.
- Peter Luschny, Lah numbers
- Peter Luschny, Partition transform
- Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See. p. 18.
- Piotr Miska and Maciej Ulas, On some properties of the number of permutations being products of pairwise disjoint d-cycles, arXiv:1904.03395 [math.NT], 2019.
- Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
- Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.
-
T := (n, k) -> `if`(n=k, 1, binomial(n-1,k-1)*n!/k!):
seq(seq(T(n, k), k=0..n), n=0..9);
-
T[n_, k_] := Binomial[n, k]*FactorialPower[n-1, n-k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)
-
@cached_function
def T(n,k):
if k<0 : return 0
if k==n: return 1
return T(n-1,k-1) + (k+n-1)*T(n-1,k)
for n in (0..8): print([T(n,k) for k in (0..n)])
A000344
a(n) = 5*binomial(2n, n-2)/(n+3).
Original entry on oeis.org
1, 5, 20, 75, 275, 1001, 3640, 13260, 48450, 177650, 653752, 2414425, 8947575, 33266625, 124062000, 463991880, 1739969550, 6541168950, 24647883000, 93078189750, 352207870014, 1335293573130, 5071418015120, 19293438101000, 73514652074500, 280531912316292
Offset: 2
G.f. = x^2 + 5*x^3 + 20*x^4 + 75*x^5 + 275*x^6 + 1001*x^7 + 3640*x^8 + ...
- C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Muniru A Asiru, Table of n, a(n) for n = 2..300(Terms 2..170 from Vincenzo Librandi)
- Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023. (Cites this sequence as "A00344")
- Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
- Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
- Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 31.
- Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.
- Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21, No. 2 (2010), pp. 265-284 (see Theorem 4.2 p. 275).
- Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
- V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
- C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146. [Annotated scanned copy]
- Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]
- Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math., Vol. 14 (1956), pp. 405-414.
- John Riordan, Letter to N. J. A. Sloane, Nov 10 1970.
- John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.
- Zoran Sunic, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, Vol. 10 (2003) Article N5.
T(n, n+5) for n=0, 1, 2, ..., array T as in
A047072.
-
List([2..30],n->5*Binomial(2*n,n-2)/(n+3)); # Muniru A Asiru, Aug 09 2018
-
[5*Binomial(2*n,n-2)/(n+3): n in [2..30]]; // Vincenzo Librandi, May 03 2011
-
A000344List := proc(m) local A, P, n; A := [1]; P := [1,1,1,1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A000344List(27); # Peter Luschny, Mar 26 2022
-
Table[5 Binomial[2n,n-2]/(n+3),{n,2,40}] (* or *) CoefficientList[Series[ (1-Sqrt[1-4 x]+x (-5+3 Sqrt[1-4 x]-(-5+Sqrt[1-4 x]) x))/(2 x^5), {x,0,38}],x] (* Harvey P. Dale, May 01 2011 *)
a[ n_] := If[ n < 0, 0, 5 Binomial[2 n, n - 2] / (n + 3)]; (* Michael Somos, May 28 2014 *)
-
a(n)=5*binomial(2*n,n-2)/(n+3) \\ Charles R Greathouse IV, Jul 25 2011
A075729
Number of different hierarchical orderings that can be formed from n labeled elements: these are divided into groups and the elements in each group are then arranged in a "preferential arrangement" or "weak order" as in A000670.
Original entry on oeis.org
1, 1, 4, 23, 173, 1602, 17575, 222497, 3188806, 50988405, 899222457, 17329515172, 362164300173, 8155216185781, 196789115887252, 5064722539020379, 138457553073641465, 4006059432756066914, 122284085809137076203, 3926775294104305483621, 132313462760902116605534
Offset: 0
a(3) = 23: Let the n = 3 individuals be named 1, 2 and 3. Let a pair of parentheses () indicate a society and let square brackets [] denote a set of disparate societies. Finally, let the ranks be ordered from left to right and separated by a colon, e.g., (1,2:3) is a society with individual 3 on top and individuals 1 and 2 on the same bottom rank.
Then the hierarchical ordering for n = 3 is composed of the following sets: [(1),(2),(3)], [(1,2)(3)], [(3,2)(1)], [(3,1)(2)], [(1:2)(3)], [(3:2)(1)], [(1:3)(2)], [(2:1)(3)], [(2:3)(1)], [(3:1)(2)], [(3:2:1)], [(1:3:2)], [(2:1:3)], [(1:2:3)], [(3:1:2)], [(2:3:1)], [(1,3:2)], [(3,2:1)], [(2,1:3)], [(3:1,2)], [(1:2,3)], [(2:3,1)], [(1,2,3)].
- Alois P. Heinz, Table of n, a(n) for n = 0..419 (first 101 terms from T. D. Noe)
- José A. Adell and Sithembele Nkonkobe, A unified generalization of Touchard and Fubini polynomial extensions, Integers (2023) 23, #A80.
- INRIA, Algolib: The Algorithms Project's Library
- P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 571.
- Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 2.
- Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
- K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem [J. Phys. A 37 (2004), 3475-3487]
- N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
- Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
- Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. - _Thomas Wieder_, Nov 14 2009
- Index entries for sequences related to parenthesizing
-
A075729 := n->n!*exp(1/4/ln(2)-3/4)/2/sqrt(Pi)/(2*ln(2))^(1/4)*exp(-n*ln(ln(2)))*exp(sqrt(2*n/ln(2)))*n^(-3/4);
with(combstruct); SetSeqSetL := [T, {T=Set(S), S=Sequence(U,card >= 1), U=Set(Z,card >=1)},labeled]; seq(count(SetSeqSetL,size=j),j=1..12);
# alternative Maple program:
b:= proc(n) option remember: `if`(n<2, 1,
(2*n-1)*b(n-1) -(n-1)*(n-2)*b(n-2))
end:
a:= n-> add(b(k)*Stirling2(n,k), k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, May 22 2018
-
Range[0, 20]!CoefficientList[Series[E^(1/(2 - E^x) - 1), {x, 0, 20}], x] (* Robert G. Wilson v, Jul 13 2004 *)
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; a[0] = 1; a[n_] := a[n] = (n-1)! Sum[a[n-k] Fubini[k, 1]/((n-k)! (k-1)!), {k, 1, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 31 2016 *)
Table[Sum[BellY[n, k, PolyLog[-Range[n], 1/2]/2], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
With[{nn=20},CoefficientList[Series[Exp[1/(2-Exp[x])-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 26 2022 *)
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a(n):= sum(sum(stirling2(n,k)*k!*binomial(k-1,m-1), k,m,n)/m!, m,1,n) /* Vladimir Kruchinin, Aug 10 2010 */
A057625
a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.
Original entry on oeis.org
1, 3, 7, 37, 121, 1201, 5041, 62161, 423361, 5473441, 39916801, 818959681, 6227020801, 130784734081, 1536517382401, 32256486662401, 355687428096001, 10679532671808001, 121645100408832001, 3770998783116364801, 59616236292028416001, 1686001119824999577601
Offset: 1
a(4) = 4! (1 + 1/2! + 1/4!) = 24 (1 + 1/2 + 1/24) = 37.
A000588
a(n) = 7*binomial(2n,n-3)/(n+4).
Original entry on oeis.org
0, 0, 0, 1, 7, 35, 154, 637, 2548, 9996, 38760, 149226, 572033, 2187185, 8351070, 31865925, 121580760, 463991880, 1771605360, 6768687870, 25880277150, 99035193894, 379300783092, 1453986335186, 5578559816632, 21422369201800, 82336410323440, 316729578421620
Offset: 0
G.f. = x^3 + 7*x^4 + 35*x^5 + 154*x^6 + 637*x^7 + 2548*x^8 + 9996*x^9 + ...
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n = 0..200
- Joerg Arndt, The a(5)=35 Young tableaux of shape [8,2].
- Dennis E. Davenport, Louis W. Shapiro, Lara K. Pudwell and Leon C. Woodson, The Boundary of Ordered Trees, J. Integer Seq., Vol. 18 (2015), Article 15.5.8; alternative link.
- Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21, No. 2 (2010), pp. 265-284 (see 4.4 p. 279).
- Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
- V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
- Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]
- Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math., Vol. 14 (1956), pp. 405-414.
- John Riordan, Letter to N. J. A. Sloane, Nov 10 1970.
- John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.
- Zoran Sunic, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, Vol. 10 (2003), Article N5.
-
a[n_] := 7*Binomial[2n, n-3]/(n + 4); Table[a[n],{n,0,27}] (* James C. McMahon, Dec 05 2023 *)
-
A000588(n)=7*binomial(2*n,n-3)/(n+4) \\ M. F. Hasler, Aug 25 2012
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my(x='x+O('x^50)); concat([0, 0, 0], Vec(x^3*((1-(1-4*x)^(1/2))/(2*x))^7)) \\ Altug Alkan, Nov 01 2015
A111884
E.g.f.: exp(x/(1+x)).
Original entry on oeis.org
1, 1, -1, 1, 1, -19, 151, -1091, 7841, -56519, 396271, -2442439, 7701409, 145269541, -4833158329, 104056218421, -2002667085119, 37109187217649, -679877731030049, 12440309297451121, -227773259993414719, 4155839606711748061, -74724654677947488521, 1293162252850914402221
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..450
- Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 4.
- Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 6.
- Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 20.
- Paul Barry, Combinatorial Polynomials as Moments, Hankel Transforms, and Exponential Riordan Arrays, J. Int. Seq. 14 (2011) 11.6.7, example 10.
- Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32.
- Aoife Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
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nn=30; CoefficientList[Series[Exp[x/(1+x)],{x,0,nn}], x] Range[0,nn]! (* Harvey P. Dale, Jul 21 2011 *)
-
A111884 = lambda n: hypergeometric([-n+1,-n], [], -1)
[Integer(A111884(n).n(100)) for n in (0..23)] # Peter Luschny, Sep 23 2014
A050351
Number of 3-level labeled linear rooted trees with n leaves.
Original entry on oeis.org
1, 1, 5, 37, 365, 4501, 66605, 1149877, 22687565, 503589781, 12420052205, 336947795317, 9972186170765, 319727684645461, 11039636939221805, 408406422098722357, 16116066766061589965, 675700891505466507541
Offset: 0
G.f. = 1 + x + 5*x^2 + 37*x^3 + 365*x^4 + 4501*x^5 + 66605*x^6 + ...
- T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.
- T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
- G. C. Greubel, Table of n, a(n) for n = 0..390
- Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 1.
- S. Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938 [math.CO], 2013.
- Marian Muresan, A concrete approach to classical analysis, CMS Books in Mathematics (2009) Table 10.2
- Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
- N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
- Index entries for sequences related to rooted trees
-
with(combstruct); SeqSeqSetL := [T, {T=Sequence(S), S=Sequence(U,card >= 1), U=Set(Z,card >=1)},labeled];
-
With[{nn=20},CoefficientList[Series[(2-E^x)/(3-2*E^x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 29 2012 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1/(2 - 1/(2 - Exp[x])), {x, 0, n}]]; (* Michael Somos, Nov 28 2014 *)
-
{a(n) = if( n<0, 0, n! * polcoeff( 1/(2 - 1/(2 - exp(x + x * O(x^n)))), n))};
-
{a(n)=if(n==0, 1, (1/6)*round(suminf(k=1, k^n * (2/3)^k *1.)))} \\ Paul D. Hanna, Nov 28 2014
-
A050351 = lambda n: sum(stirling_number2(n,k)*(2^(k-1))*factorial(k) for k in (0..n)) if n>0 else 1
[A050351(n) for n in (0..17)] # Peter Luschny, Jan 18 2016
A082579
Expansion of e.g.f.: exp( x/(1-x)^2 ).
Original entry on oeis.org
1, 1, 5, 31, 241, 2261, 24781, 309835, 4342241, 67308841, 1141960501, 21026890391, 417264626065, 8871853115581, 201100863674621, 4838817223845571, 123128720142540481, 3302478863343928145, 93091427773284348901, 2750635764338982054031, 84994418675445218025521
Offset: 0
-
A082579:= func< n | n eq 0 select 1 else (&+[Factorial(n)*Binomial(n+k-1, n-k)/Factorial(k): k in [1..n]]) >;
[A082579(n): n in [0..25]]; // G. C. Greubel, Feb 23 2021
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nn=20;Range[0,nn]!CoefficientList[Series[Exp[ x/(1-x)^2],{x,0,nn}],x] (* Geoffrey Critzer, Nov 04 2012 *)
nn = 20; Range[0, nn]! * CoefficientList[Series[Product[Exp[k*x^k], {k, 1, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 21 2016 *)
Table[If[n==0, 1, n*n!*HypergeometricPFQ[{1-n, n+1}, {3/2, 2}, -1/4]], {n, 0, 25}] (* G. C. Greubel, Feb 23 2021 *)
-
a(n):=n!*sum(binomial(n+k-1,2*k-1)/k!,k,1,n); /* Vladimir Kruchinin, Apr 21 2011 */
-
my(x='x+O('x^33));
Vec(serlaplace(exp( x/(1-x)^2 )))
/* Joerg Arndt, Sep 14 2012 */
-
[1 if n==0 else factorial(n)*sum( binomial(n+k-1, n-k)/factorial(k) for k in (1..n)) for n in (0..25)] # G. C. Greubel, Feb 23 2021
A002872
Number of partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles.
Original entry on oeis.org
1, 2, 7, 31, 164, 999, 6841, 51790, 428131, 3827967, 36738144, 376118747, 4086419601, 46910207114, 566845074703, 7186474088735, 95318816501420, 1319330556537631, 19013488408858761, 284724852032757686, 4422344774431494155, 71125541977466879231
Offset: 0
For a(2)=7, the row patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD. The loop patterns are AAAA, AAAB, AABB, AABC, ABAB, ABAC, and ABCD. - _Robert A. Russell_, Apr 24 2018
- D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 765). - Robert A. Russell, Apr 28 2018
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Alois P. Heinz, Table of n, a(n) for n = 0..513 (first 101 terms from T. D. Noe)
- Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 12, 18, 29.
- T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv preprint arXiv:1302.6150 [math.RT], 2013.
- Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
- Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
- T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
- J. Pasukonis and S. Ramgoolam, From counting to construction for BPS states in N=4 SYM, arXiv:1010.1683 [hep-th], 2010, (E.2).
- J. Pasukonis and S. Ramgoolam, From counting to construction for BPS states in N=4 SYM, J. High En. Phys. 2011 (2) (2011), (E.2).
- OEIS Wiki, Sorting numbers
- R. Orellana and M. Zabrocki, Symmetric group characters as symmetric functions, arXiv preprint arXiv:1605.06672 [math.CO], 2016-2017.
- J. Quaintance, Letter representations of rectangular m x n x p proper arrays, arXiv:math/0412244 [math.CO], 2004-2006.
- Index entries for sequences related to sorting
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a:= proc(n) option remember; `if`(n=0, 1, add((1+
2^(j-1))*binomial(n-1, j-1)*a(n-j), j=1..n))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Oct 29 2015
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u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; Table[u[n,2],{n,0,12}] (* Wouter Meeussen, Dec 06 2008 *)
mx = 16; p = 2; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Aeven[m_, k_] := Aeven[m, k] = If[m>0, k Aeven[m-1, k] + Aeven[m-1, k-1]
+ Aeven[m-1, k-2], Boole[m==0 && k==0]]
Table[Sum[Aeven[m, k], {k, 0, 2m}], {m, 0, 30}] (* Robert A. Russell, Apr 24 2018 *)
x[n_] := x[n] = If[n<2, n+1, 2x[n-1] + (n-1)x[n-2]]; (* A005425 *)
Table[Sum[StirlingS2[n, k] x[k], {k, 0, n}], {n, 0, 20}] (* Robert A. Russell, Apr 28 2018, from Knuth reference *)
Table[Sum[Binomial[n,k] * 2^k * BellB[k, 1/2] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)
A001859
Triangular numbers plus quarter-squares: n*(n+1)/2 + floor((n+1)^2/4) (i.e., A000217(n) + A002620(n+1)).
Original entry on oeis.org
0, 2, 5, 10, 16, 24, 33, 44, 56, 70, 85, 102, 120, 140, 161, 184, 208, 234, 261, 290, 320, 352, 385, 420, 456, 494, 533, 574, 616, 660, 705, 752, 800, 850, 901, 954, 1008, 1064, 1121, 1180, 1240, 1302, 1365, 1430, 1496, 1564, 1633, 1704, 1776, 1850, 1925
Offset: 0
For n=1 we find 2 planted trees with 8 nodes, 3 of which are internal (i) and 5 are endpoints (e):
.e...e...e...e....e...e....
...i.......i........i...e..
.......i..............i...e
.......e................i..
........................e..
G.f. = 2*x + 5*x^2 + 10*x^3 + 16*x^4 + 24*x^5 + 33*x^6 + 44*x^7 + 56*x^8 + ...
- John Riordan, personal communication.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n = 0..1000
- D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.3
- S. V. Gervacio and H. Maehara, Partial order on a family of k-subsets of a linearly ordered set, Discr. Math., 306 (2006), 413-419.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- J. Riordan, Letter to N. J. A. Sloane, Oct. 1970
- S. G. Wagner, An identity for the cycle indices of rooted tree automorphism groups, Elec. J. Combinat., 13 (2006), #R00.
- Index entries for sequences related to rooted trees.
- Index entries for sequences related to trees.
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Antidiagonal sums of array
A003984.
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a001859 n = a000217 n + a002620 (n + 1) -- Reinhard Zumkeller, Dec 20 2012
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A001859:=(-1-z^2-2*z^3+z^4)/(z+1)/(z-1)^3; # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence with an additional leading 1
with (combinat):seq(count(Partition((3*n+2)), size=3), n=0..50); # Zerinvary Lajos, Mar 28 2008
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With[{nn=60},Total/@Thread[{Accumulate[Range[0,nn]],Floor[Range[ nn+1]^2/4]}]] (* or *) LinearRecurrence[{2,0,-2,1},{0,2,5,10},60] (* Harvey P. Dale, Apr 01 2012 *)
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{a(n) = n + (3*n^2 + 1) \ 4};
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