A008977 -id:A008977 - cp's OEIS frontend

A269122 Number of sequences with n copies each of 1,2,3,4 avoiding the pattern 1234.

1, 23, 1879, 173891, 16140983, 1474050783, 132279480115, 11690616534627, 1020149742404727, 88094027572412735, 7541872062618216079, 641058122723345275763, 54163773797158984485203, 4553249681522845632696611, 381122944171897823109474291

Offset: 0

Alois P. Heinz, Feb 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..525

Row n=4 of A269129.

Cf. A008977, A268840.

a(n) = A008977(n) - A268840(n).

A377218 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(24n) = (4n)!/n!^4 for n >= 0.

Original entry on oeis.org

1, 1, 29, 2246, 239500, 30318701, 4271201506, 647359627557, 103476937050223, 17223017775652625, 2959285397777331751, 521687007046376376544, 93932798602803741121051, 17215649571517858590782737, 3203146941738318544432065500, 603763082812549420389330837978, 115095760617137117019641563685386

Offset: 0

Peter Bala, Oct 20 2024

Compare with A000984(n) = [x^n] (1 + x)^(2*n) = (2*n)!/n!^2.
The central binomial coefficients A000984(n) satisfy the supercongruences u(n*p^k) == u(n*p^(k-1)) (mod p^(3*k)) for all primes p >= 5 and positive integers n and k.
More generally, for positive integers r and s, the sequence {u(r,s; n) : n >= 0} defined by u(r,s; n) = [x^(s*n)] (1 + x)^(r*n) = binomial(r*n, s*n) satisfies the same supercongruences (Meštrović, Section 6, equation 39).
Conjecture: for positive integers r and s, the sequence {v(r,s; n) : n >= 0} defined by v(r,s; n) = [x^(s*n)] A(x)^(r*n) also satisfies the same supercongruences.

Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], (2011).

Cf. A008977, A082368, A333042, A370294, A377217, A377219.

Order := 25:
E(x) := exp(add((4*n)!/n!^4 * x^n/n, n = 1..25)):
solve(series(x*E(x),x) = y, x):
convert(%, polynom):
g := taylor(y/%, y = 0, 25):
seq(coeftayl(g^(1/24), y = 0,  n), n = 0..20);

O.g.f.: A(x) = ( x/(x * series_reversion(E(x)))^(1/24), where E(x) = exp(Sum_{n >= 1} (4*n)!/n!^4 *x^n/n) is the o.g.f. of A333042.

A059068 Card-matching numbers (Dinner-Diner matching numbers).

Original entry on oeis.org

1, 9, 8, 6, 0, 1, 297, 672, 736, 480, 246, 64, 24, 0, 1, 13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1, 748521, 3662976, 8607744, 12880512, 13731616, 11042688, 6928704, 3458432, 1395126

Offset: 0

Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

This is a triangle of card matching numbers. A deck has 4 kinds of cards, n of each kind. The deck is shuffled and dealt in to 4 hands with each with n cards. A match occurs for every card in the j-th hand of kind j. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..4n). The probability of exactly k matches is T(n,k)/((4n)!/n!^4).
Rows have lengths 1,5,9,13,...
Analogous to A008290 - Zerinvary Lajos, Jun 22 2005

			There are 736 ways of matching exactly 2 cards when there are 2 cards of each kind and 4 kinds of card so T(2,2)=736.
Triangle begins:
      1;
      9,     8,     6,     0,     1;
    297,   672,   736,   480,   246,    64,    24,    0,    1;
  13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1;
  ...

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.

F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
Barbara H. Margolius, Dinner-Diner Matching Probabilities
B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620.
Index entries for sequences related to card matching

Cf. A008290, A059056-A059071.
Cf. A008290.
Row sums give A008977.

p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k);
for n from 0 to 5 do seq(coeff(f(t,4,n),t,m)/n!^4,m=0..4*n); od;

p[x_, k_] := k!^2*Sum[x^j/((k-j)!^2*j!), {j, 0, k}]; r[x_, n_, k_] := p[x, k]^n; f[t_, n_, k_] := Sum[ Coefficient[r[x, n, k], x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Table[ Coefficient[f[t, 4, n], t, m]/n!^4, {n, 0, 4}, {m, 0, 4*n}] // Flatten (* Jean-François Alcover, Dec 17 2012, translated from Maple *)

G.f.: sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k) where n is the number of kinds of cards (4 in this case), k is the number of cards of each kind and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*sum(x^j/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.

A248600 G.f.: Sum_{n>=0} R_n(x+xy) x^(2n)y^n / (1-x-xy)^(4n+1) = Sum_{n>=0} Sum_{k=0..n} C(n,k)^4 * x^ny^k, where R_n(x+xy) equals the n-th row polynomial R_n(z) = Sum_{k=0..2n} T(n,k)z^k at z = x+x*y.

Original entry on oeis.org

1, 14, 8, 2, 786, 1056, 576, 96, 6, 61340, 131760, 117900, 48320, 9540, 720, 20, 5562130, 16481920, 20917120, 13847680, 5118400, 1025920, 105280, 4480, 70, 549676764, 2079579600, 3444581700, 3165926400, 1755532800, 598123008, 123656400, 14716800, 926100, 25200, 252, 57440496036

Offset: 0

Paul D. Hanna, Oct 11 2014

			Triangle begins:
[1],
[14, 8, 2],
[786, 1056, 576, 96, 6],
[61340, 131760, 117900, 48320, 9540, 720, 20],
[5562130, 16481920, 20917120, 13847680, 5118400, 1025920, 105280, 4480, 70],
[549676764, 2079579600, 3444581700, 3165926400, 1755532800, 598123008, 123656400, 14716800, 926100, 25200, 252],
[57440496036, 264565490112, 542687590368, 640299696960, 477284304420, 233110386432, 75243589344, 15835792896, 2103157980, 165802560, 7051968, 133056, 924],
[6242164112184, 33895475918304, 83073660613944, 119912994225024, 112698387745944, 72172565713248, 32111980788888, 9951304416768, 2124873478728, 305035899168, 28270554312, 1584815232, 48600552, 672672, 3432],
[698300344311570, 4368053451041280, 12465205610457600, 21305587665922560, 24216302627637120, 19255941998092800, 10989839486545920, 4550117424652800, 1366687981264320, 295074717949440, 44954858108160, 4691645038080, 320878958400, 13445752320, 311351040, 3294720, 12870], ...
where this triangle forms the coefficients in the series
B(x,y) = 1/(1-x-x*y) +
(14 + 8*(x+x*y) + 2*(x+x*y)^2) * x^2*y/(1-x-x*y)^5 +
(786 + 1056*(x+x*y) + 576*(x+x*y)^2 + 96*(x+x*y)^3 + 6*(x+x*y)^4) * x^4*y^2/(1-x-x*y)^9 +
(61340 + 131760*(x+x*y) + 117900*(x+x*y)^2 + 48320*(x+x*y)^3 + 9540*(x+x*y)^4 + 720*(x+x*y)^5 + 20*(x+x*y)^6) * x^6*y^3/(1-x-x*y)^13 +...
such that the sum may be expressed using binomial coefficients C(n,k)^4 like so:
B(x,y) =  1 +
x*(1 + y) +
x^2*(1 + 2^4*y + y^2) +
x^3*(1 + 3^4*y + 3^4*y^2 + y^3) +
x^4*(1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4) +
x^5*(1 + 5^4*y + 10^4*y^2 + 10^4*y^3 + 5^4*y^4 + y^5) +
x^6*(1 + 6^4*y + 15^4*y^2 + 20^4*y^3 + 15^4*y^4 + 6^4*y^5 + y^6) +...
The central terms of the rows begin:
[1, 8, 576, 48320, 5118400, 598123008, 75243589344, 9951304416768, 1366687981264320, ...].

Cf. A050983, A000984, A008977, A002897, A187056, A248706.

Leftmost border equals A050983, de Bruijn's S(4,n):
T(n,0) = Sum_{k=0..2*n} (-1)^(n+k) * C(2*n,k)^4.
Rightmost border equals A000984, the central binomial coefficients:
T(n,2*n) = Sum_{k=0..2*n} (-1)^(n+k)* C(2*n,k)^2 = (2*n)!/(n!)^2.
Row sums equal A008977(n) = (4*n)!/(n!)^4.
Sum_{k=0..n} (-1)^k * T(n,k) = A002897(n) = C(2*n,n)^3.

A307618 A Calabi-Yau period integral: a(n) = C(4n,2n)C(2n,n)^3.

Original entry on oeis.org

1, 48, 15120, 7392000, 4414410000, 2956651746048, 2133278987583744, 1621682968820428800, 1281351259836532170000, 1043032815185819858400000, 869343653096068540955685120, 738637974389826550020188712960, 637665137404661719206664998969600

Offset: 0

Bradley Klee, Jun 04 2019

nonn

Entry number six in the "Big Table" of Almkvist et al. (see links). The period T(x) = Sum_{n>=0} a(n)*x^(2*n) is also the first x-derivative of the 6-volume associated to the algebraic variety V6 = P1 & P2 & P3, with P1 : X1^2 + Y1^2 = X2^2 + Y2^2, P2 : X2^2 + Y2^2 = X3^2 + Y3^2, P3 : x=(X1^2 + X2^2 + X3^2 + Y1^2 + Y2^2 + Y3^2)^3*(1 - X1*X2*X3*Y1*Y2*Y3). The small x limit reduces V6 to a 6-ball with 6-volume proportional to x. Similar constructions are known to exist for a few other geometries on Almkvist's list, most notably #3: A186420, and #16: A039699.

G. Almkvist et al., Tables of Calabi-Yau Equations, arXiv:math/0507430 [math.AG], 2005-2010, p. 10.
G. Almkvist et al., Raw Calabi-Yau Period Data, Johannes Gutenberg Universität Mainz, 2019, case 1.6.

Hadamard Factors: A000984, A002894, A002897, A001448, A000897, A008977.

Calabi-Yau Periods: A008978, A186420, A268553, A039699.

Binomial[4*#,2*#]*Binomial[2*#,#]^3&/@Range[0,10]

G.f.: 4F3({1/4, 3/4, 1/2, 1/2}, {1, 1, 1}, 1024*x).
Define the period integral:
dt(x) = dz1*dz2*dz3/sqrt(1-32*x*cos(z1)*cos(z2)*cos(z3)).
T(x)=1/(2*Pi)^3*Integral_{0..2*Pi,0..2*Pi,0..2*Pi} dt(x),
the Picard-Fuchs coefficients:(c0,c1,c2,c3,c4)=
(768*x, 14592*x^2-1, x*(25344*x^2-7), 2*x^2*(5120*x^2-3), x^3*(32*x-1)*(32*x+1)),
and the certificate function:
G(z1,z2,z3)=(16*sin(z1)*(
       48*x*cos(z1)
     + cos(z2)*cos(z3)
     + 48*x*cos(z1)*(cos(z3)^2 + cos(z2)^2)
     + 2304*x^2*cos(z1)^2*cos(z2)*cos(z3)
     + 80*x*cos(z1)*cos(z2)^2*cos(z3)^2
     + 384*x^2*cos(z1)^2*(cos(z2)*cos(z3)^3 + cos(z2)^3*cos(z3))
     + 256*x^2*cos(z1)^2*cos(z2)^3*cos(z3)^3)
    )/(3*(1 - 32*x*cos(z1)*cos(z2)*cos(z3))^(7/2)),
Then: 0 = Sum_{n=0..4}cn*d^n/dx^n dt(x) + d/dz1 G(z1,z2,z3) + d/dz2 G(z2,z3,z1) + d/dz3 G(z3,z1,z2), thus: 0 = Sum_{n=0..4} cn*d^n/dx^n T(x).
Furthermore, let (a1,a2,a3)=(c1,c2,c3)/c0, then also: 0 = (1/2)*a2*a3 - (1/8)*a3^3 + d/dx(a2) - (3/4)*a3*d/dx(a3) - (1/2)*d^2/dx^2(a3) - a1.
D-finite with recurrence: n^4*a(n) -16*(4*n-1)*(4*n-3)*(-1+2*n)^2*a(n-1)=0. - R. J. Mathar, Jan 27 2020

A318105 Triangle read by rows: T(n,k) = (4n - 3k)!/((n-k)!^4*k!).

Original entry on oeis.org

1, 24, 1, 2520, 120, 1, 369600, 22680, 360, 1, 63063000, 4804800, 113400, 840, 1, 11732745024, 1072071000, 33633600, 415800, 1680, 1, 2308743493056, 246387645504, 9648639000, 168168000, 1247400, 3024, 1, 472518347558400, 57718587326400, 2710264100544, 61108047000, 672672000, 3243240, 5040, 1

Offset: 0

Gheorghe Coserea, Oct 15 2018

Diagonal of rational function R(x,y,z,w,t) = 1/(1 - (x+y+z+w + t*x*y*z*w)) with respect to x,y,z,w, i.e., T(n,k) = [(xyzw)^n*t^k] R(x,y,z,w,t).

Annihilating differential operator: x^2*(3*t*x + 1)^2*((t*x - 1)^4 - 256*x)*Dx^3 + 3*x*(3*t*x + 1)*((t*x - 1)^3*(6*t^2*x^2 + 3*t*x - 1) - 384*x*(t*x + 1))*Dx^2 + (t*x - 1)*((t*x - 1)*(63*t^4*x^4 + 66*t^3*x^3 - 18*t*x + 1) + 48*x*(15*t*x + 17))*Dx + (t*x - 1)*(t*(9*t^4*x^4 + 12*t^3*x^3 + 6*t^2*x^2 - 12*t*x + 1) - 24*(15*t*x - 1)).

			A(x;t) = 1 + (24 + t)*x + (2520 + 120*t + t^2)*x^2 + (369600 + 22680*t + 360*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k [0]            [1]           [2]         [3]        [4]      [5]   [6]
[0] 1;
[1] 24,            1;
[2] 2520,          120,          1;
[3] 369600,        22680,        360,        1;
[4] 63063000,      4804800,      113400,     840,       1;
[5] 11732745024,   1072071000,   33633600,   415800,    1680,    1;
[6] 2308743493056, 246387645504, 9648639000, 168168000, 1247400, 3024, 1;
[7] ...

Gheorghe Coserea, Rows n=0..100, flattened

Cf. A008977, A082488, A125143, A318107.

t[n_,k_] := (4*n - 3*k)!/((n-k)!^4*k!); Table[t[n, k], {n, 0, 10}, {k , 0,n}] // Flatten  (* Amiram Eldar, Nov 07 2018 *)

T(n, k) = (4*n-3*k)!/(k!*(n-k)!^4);
concat(vector(8, n, vector(n, k, T(n-1, k-1))))
/*
test:
P(n, v='t) = subst(Polrev(vector(n+1, k, T(n, k-1)), 't), 't, v);
diag(expr, N=22, var=variables(expr)) = {
  my(a = vector(N));
  for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
  for (n = 1, N, a[n] = expr;
    for (k = 1, #var, a[n] = polcoef(a[n], n-1)));
  return(a);
};
apply_diffop(p, s) = {
  s=intformal(s);
  sum(n=0, poldegree(p, 'Dx), s=s'; polcoef(p, n, 'Dx) * s);
};
\\ diagonal property:
x='x; y='y; z='z; w='w; t='t;
diag(1/(1 - (x+y+z+w + t*x*y*z*w)), 9, [x, y, z, w]) == vector(9, n, P(n-1))
\\ annihilating diffop:
y = Ser(vector(101, n, P(n-1)), 'x);
p = x^2*(3*t*x + 1)^2*((t*x - 1)^4 - 256*x)*Dx^3 + 3*x*(3*t*x + 1)*((t*x - 1)^3*(6*t^2*x^2 + 3*t*x - 1) - 384*x*(t*x + 1))*Dx^2 + (t*x - 1)*((t*x - 1)*(63*t^4*x^4 + 66*t^3*x^3 - 18*t*x + 1) + 48*x*(15*t*x + 17))*Dx + (t*x - 1)*(t*(9*t^4*x^4 + 12*t^3*x^3 + 6*t^2*x^2 - 12*t*x + 1) - 24*(15*t*x - 1));
0 == apply_diffop(p, y)
*/

Let P_n(t) = Sum_{k=0..n} T(n,k)*t^k. Then A125143(n) = P_n(-27), A008977(n) = P_n(0), A082488(n) = P_n(1).

A324469 Exponent of highest power of 3 that divides multinomial(4*n;n,n,n,n).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 2, 5, 6, 1, 2, 3, 2, 3, 6, 4, 6, 7, 2, 3, 4, 5, 6, 8, 6, 8, 9, 1, 2, 3, 2, 3, 5, 3, 6, 7, 2, 3, 4, 3, 4, 8, 6, 8, 9, 4, 5, 6, 6, 7, 9, 7, 9, 10, 2, 3, 4, 3, 4, 6, 4, 9, 10, 5, 6, 7, 6, 7, 10, 8, 10, 11, 6, 7, 8, 8, 9, 11, 9, 11, 12, 1, 2

Offset: 0

N. J. A. Sloane, Mar 03 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Analogs for binomial and trinomials: A000989, A053735. See also A324467.

Cf. A007949 (3-adic valuation of n), A008977.

[seq(padic[ordp](combinat[multinomial](4*n, n$4), 3), n=0..128)];

s[n_] := Plus @@ IntegerDigits[n, 3]; a[n_] := 2*s[n] - s[4*n]/2; Array[a, 100, 0] (* Amiram Eldar, Feb 21 2021 *)

a(n) = 2*A000989(n) + A000989(2*n). - Charlie Neder, Mar 09 2019
From Amiram Eldar, Feb 21 2021: (Start)
a(n) = A007949(A008977(n)).
a(n) = 2*A053735(n) - A053735(4*n)/2. (End)

A342107 a(n) = Sum_{k=0..n} (4*k)!/k!^4.

Original entry on oeis.org

1, 25, 2545, 372145, 63435145, 11796180169, 2320539673225, 474838887231625, 100035931337622625, 21552788197602942625, 4726913659271173170145, 1051798742538350304851425, 236861100204680963085573025

Offset: 0

Seiichi Manyama, Feb 28 2021

Partial sums of A008977.

In general, for m > 1, Sum_{k=0..n} (m*k)!/k!^m ~ m^(m*n + m + 1/2) / ((m^m - 1) * (2*Pi*n)^((m-1)/2)). - Vaclav Kotesovec, Feb 28 2021

Cf. A006134, A008977, A188441, A221177.

A342107 := proc(n)
    add((4*k)!/k!^4,k=0..n) ;
end proc:
seq(A342107(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

Table[Sum[(4*k)!/k!^4, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 28 2021 *)

PARI
```
a(n) = sum(k=0, n, (4*k)!/k!^4);
```

a(n) ~ 2^(8*n + 15/2) / (255 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Feb 28 2021

D-finite with recurrence n^3*a(n) +(-257*n^3+384*n^2-176*n+24)*a(n-1) +8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Dec 04 2023

cp's OEIS Frontend

A269122 Number of sequences with n copies each of 1,2,3,4 avoiding the pattern 1234.

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A377218 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(24*n) = (4*n)!/n!^4 for n >= 0.

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A059068 Card-matching numbers (Dinner-Diner matching numbers).

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A248600 G.f.: Sum_{n>=0} R_n(x+x*y) * x^(2*n)*y^n / (1-x-x*y)^(4*n+1) = Sum_{n>=0} Sum_{k=0..n} C(n,k)^4 * x^n*y^k, where R_n(x+x*y) equals the n-th row polynomial R_n(z) = Sum_{k=0..2*n} T(n,k)*z^k at z = x+x*y.

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A307618 A Calabi-Yau period integral: a(n) = C(4*n,2*n)*C(2*n,n)^3.

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A318105 Triangle read by rows: T(n,k) = (4*n - 3*k)!/((n-k)!^4*k!).

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A324469 Exponent of highest power of 3 that divides multinomial(4*n;n,n,n,n).

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A342107 a(n) = Sum_{k=0..n} (4*k)!/k!^4.

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A377218 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(24n) = (4n)!/n!^4 for n >= 0.

A248600 G.f.: Sum_{n>=0} R_n(x+xy) x^(2n)y^n / (1-x-xy)^(4n+1) = Sum_{n>=0} Sum_{k=0..n} C(n,k)^4 * x^ny^k, where R_n(x+xy) equals the n-th row polynomial R_n(z) = Sum_{k=0..2n} T(n,k)z^k at z = x+x*y.

A307618 A Calabi-Yau period integral: a(n) = C(4n,2n)C(2n,n)^3.

A318105 Triangle read by rows: T(n,k) = (4n - 3k)!/((n-k)!^4*k!).