A000707 -id:A000707 - cp's OEIS frontend

A005288 a(n) = C(n,5) + C(n,4) - C(n,3) + 1, n >= 7.

3, 22, 71, 169, 343, 628, 1068, 1717, 2640, 3914, 5629, 7889, 10813, 14536, 19210, 25005, 32110, 40734, 51107, 63481, 78131, 95356, 115480, 138853, 165852, 196882, 232377, 272801, 318649, 370448, 428758, 494173, 567322

Offset: 6

N. J. A. Sloane

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 15.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241-242. (Annotated scanned copy)
R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.
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
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A008302.

Join[{3},Table[Binomial[n,5]+Binomial[n,4]-Binomial[n,3]+1,{n,7,50}]] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{3,22,71,169,343,628,1068},50] (* Harvey P. Dale, Aug 30 2021 *)

a(n) = C(n+3, 5) - C(n+2, 3) + C(n, 0).
G.f.: 3*x^6 -x^7*(x-2)*(2*x^4-11*x^3+24*x^2-25*x+11)/(x-1)^6. Simon Plouffe in his 1992 dissertation
a(n) = (n+4)*(n-3)*(n^3-6*n^2+3*n-10)/120, n >= 7. - R. J. Mathar, May 19 2013

A005287 Number of permutations of [n] with four inversions.

Original entry on oeis.org

5, 20, 49, 98, 174, 285, 440, 649, 923, 1274, 1715, 2260, 2924, 3723, 4674, 5795, 7105, 8624, 10373, 12374, 14650, 17225, 20124, 23373, 26999, 31030, 35495, 40424, 45848, 51799, 58310, 65415, 73149, 81548, 90649, 100490, 111110, 122549, 134848, 148049

Offset: 4

N. J. A. Sloane, Robert G. Wilson v

			[2, 4, 3, 1], [3, 2, 4, 1], [3, 4, 1, 2], [4, 1, 3, 2], [4, 2, 1, 3] have 4 inversions.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 255, #2, b(n,4).
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
R. K. Guy, personal communication.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Exercise 1.30, p. 49.

Vincenzo Librandi, Table of n, a(n) for n = 4..10000
R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.
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
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A008302, A005286, A005288.

[n*(n+1)*(n^2+n-14)/24: n in [4..50]]; // Vincenzo Librandi, Jul 17 2011

[seq(binomial(n,4)+binomial(n,3)-binomial(n,2), n=5..43)]; # Zerinvary Lajos, Jul 23 2006

CoefficientList[Series[(z^4 - 3*z^3 + z^2 + 5*z - 5)/(z - 1)^5, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 16 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{5,20,49,98,174},40] (* Harvey P. Dale, Aug 25 2016 *)

PARI
```
a(n)=if(n<4,0,n*(n+1)*(n^2+n-14)/24)
```

a(n) = n*(n+1)*(n^2+n-14)/24.
G.f.: x^4*(-5 + 5*x + x^2 - 3*x^3 + x^4) / (x-1)^5. - Simon Plouffe in his 1992 dissertation
a(n) = binomial(n+1,4) + binomial(n+1,3) - binomial(n+1,2). - Zerinvary Lajos, Jul 23 2006

A022579 Expansion of Product_{m>=1} (1+x^m)^14.

Original entry on oeis.org

1, 14, 105, 574, 2576, 10052, 35273, 113794, 342699, 974176, 2635955, 6833540, 17061345, 41197422, 96544003, 220212384, 490104727, 1066552228, 2273590095, 4755188704, 9771319068, 19751596934, 39317784863, 77150246040, 149357609184, 285497384004, 539227765104, 1006978117880

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Column k=14 of A286335. Cf. A000707, A023003.

Coefficients(&*[(1+x^m)^14:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

nmax=50; CoefficientList[Series[Product[(1+q^m)^14,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^14)) \\ G. C. Greubel, Feb 25 2018

a(n) ~ (7/6)^(1/4) * exp(Pi * sqrt(14*n/3)) / (256 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (14/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f. A(x) = (1/2)*( G(sqrt(x)) + G(-sqrt(x)) )/G(x^4), where G(x) = Product_{n >= 1} 1/(1 - x^n)^4 is the g.f. of A023003 (see also A000727). - Peter Bala, Oct 05 2023

A060701 Table by antidiagonals of Mahonian numbers T(n,k): permutations of n letters with k inversions.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 1, 5, 4, 1, 0, 0, 0, 6, 9, 5, 1, 0, 0, 0, 5, 15, 14, 6, 1, 0, 0, 0, 3, 20, 29, 20, 7, 1, 0, 0, 0, 1, 22, 49, 49, 27, 8, 1, 0, 0, 0, 0, 20, 71, 98, 76, 35, 9, 1, 0, 0, 0, 0, 15, 90, 169, 174, 111, 44, 10, 1, 0, 0, 0, 0, 9, 101, 259, 343, 285

Offset: 0

Henry Bottomley, Apr 25 2001

			1;
0,1;
0,1,1;
0,0,2,1;
0,0,2,3,1;
0,0,1,5,4,1;
0,0,0,6,9,5,1; ...
[1, 4, 2, 3], [1, 3, 4, 2], [2, 1, 4, 3], [2, 3, 1, 4], [3, 1, 2, 4] have 2 inversions so T(4, 2)=5.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Corollary 1.3.10, p. 21.

A008302 is the main entry for these numbers. Columns include A000012, A000027, A000096, A005286, A005287, A005288. Diagonals include A000707, A001892, A001893, A001894, A005283, A005284, A005285.

T(n,k)=polcoeff(prod(j=1,n-1,sum(i=0,j,x^i)),k)

T(n, k)=sum_{j=0..n}[T(n-1, k-j)].

Product (1+x+...+x^k), k=1..n-1 = Sum T(n, k)x^k, k=0..n(n-1)/2.

Additional comments from Michael Somos, Jun 23 2002.

A129481 a(n) = coefficient of x^n in n!*Product_{k=0..n} [Sum_{j=0..k} x^j/j! ].

Original entry on oeis.org

1, 1, 3, 19, 175, 2111, 31321, 550810, 11194177, 258068893, 6653230111, 189653427206, 5922604033567, 201075967613262, 7373834652641003, 290474615891145106, 12232735359488840833, 548429151685677131389

Offset: 0

Paul D. Hanna, Apr 17 2007

nonn

a(n) is also the number of ordered submultisets of A000707. - J. M. Bergot, Aug 13 2016

			a(2) = [x^2] 2!*(1)*(1+x)*(1+x+x^2/2!) = [x^2] (2 +4*x +3*x^2 +x^3) = 3.
a(3) = [x^3] 3!*(1)*(1+x)*(1 + x + x^2/2!)*(1 + x + x^2/2! + x^3/3!) =
[x^3] (6 + 18*x + 24*x^2 + 19*x^3 +...) = 19.

Vaclav Kotesovec, Table of n, a(n) for n = 0..150

Cf. A000707.

m:=30; R:=PowerSeriesRing(Integers(), m+2);
p:= func< n,x | (&*[ (&+[x^j/Factorial(j): j in [0..k]]) : k in [0..n]]) >;
A129481:= func< n | Coefficient(R!(Laplace( p(n,x) )), n) >;
[A129481(n): n in [0..m]]; // G. C. Greubel, Feb 12 2024

Flatten[{1,Table[Coefficient[Expand[n!*Product[Sum[x^j/j!,{j,0,k}],{k,0,n}]],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)

{a(n)=n!*polcoeff(prod(k=0,n,sum(j=0,k,x^j/j!)+x*O(x^n)),n)}

def p(n,x): return product(sum(x^j/factorial(j) for j in range(k+1)) for k in range(n+1))
def A129481(n): return factorial(n)*( p(n,x) ).series(x, 101).list()[n]
[A129481(n) for n in range(31)] # G. C. Greubel, Feb 13 2024

a(n) ~ c * n^n, where c = 0.660942456683588459181273625114230472913... . - Vaclav Kotesovec, Feb 10 2015

A382454 Number of solutions winning the Tchoukaillon game with n seeds and 2n pits.

Original entry on oeis.org

1, 2, 9, 49, 285, 1717, 10569, 66013, 416687, 2651355, 16976806, 109256095, 706071989, 4579020513, 29784426945, 194231327451, 1269457354069, 8313189986612, 54534379879411, 358298017624625, 2357331709694072, 15528887031395023, 102412421113465576, 676104332189192702

Offset: 0

Darío Clavijo, May 26 2025

nonn

a(n) is the number of permutations of [2n+1] with n inversions. a(2) = 9: 12453, 12534, 13254, 13425, 14235, 21354, 21435, 23145, 31245. - Alois P. Heinz, May 27 2025

Mancala World, Tchoukaillon

Cf. A000707, A008302.

a:= n-> coeff(series(mul((1-q^j)/(1-q), j=1..2*n+1), q, n+1), q, n):
seq(a(n), n=0..23);  # Alois P. Heinz, May 27 2025

a[n_]:=Coefficient[Series[Product[(1-q^j)/(1-q),{j,1,2*n+1}],{q,0,n+1}]//Normal,q,n];Array[a, 24, 0] (* Shenghui Yang, Jun 02 2025 *)

def a(n):
    if n == 0: return 1
    p = [1]
    for j in range(1, (n << 1) + 2):
        np = [0] * (len(p) + j - 1)
        for k in range(len(p)):
            for l in range(j):
                if (kl:=k+l) <= n:
                    np[kl] += p[k]
        p = np[:n+1]
    return p[n]
print([a(n) for n in range(1,24)])

a(n) = T(2n,n) with T(x,y) = Sum_{v=0..min(x,y)} T(x-1, y-v) and T(0,y) = 1 if y = 0 else 0.

a(n) = A008302(2n+1,n).

cp's OEIS Frontend

A005288 a(n) = C(n,5) + C(n,4) - C(n,3) + 1, n >= 7.

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A005287 Number of permutations of [n] with four inversions.

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A022579 Expansion of Product_{m>=1} (1+x^m)^14.

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A060701 Table by antidiagonals of Mahonian numbers T(n,k): permutations of n letters with k inversions.

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A129481 a(n) = coefficient of x^n in n!*Product_{k=0..n} [Sum_{j=0..k} x^j/j! ].

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A382454 Number of solutions winning the Tchoukaillon game with n seeds and 2n pits.

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Maple

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