A176730 -id:A176730 - cp's OEIS frontend

A117226 Number of permutations of [n] avoiding the consecutive pattern 1243.

1, 1, 2, 6, 23, 110, 630, 4204, 32054, 274914, 2619692, 27459344, 313990182, 3889585408, 51888955808, 741668212080, 11307669002720, 183174676857608, 3141820432768752, 56882461258572976, 1084056190235653304, 21692744773505849952, 454758269790599361968

Offset: 0

Steven Finch, Apr 26 2006

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 1243. It is the same as the number of permutations which avoid 3421, 4312 or 2134.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 0..60 from Ray Chandler)
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes, 2011.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see p. 120.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, arXiv:math/0505254 [math.CO], 2005.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006), 138-155.
Steven Finch, Pattern-Avoiding Permutations. [Archived copy]
Steven Finch, Pattern-Avoiding Permutations. [Cached copy, with permission]
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Wikipedia, Falling and rising factorials.

Cf. A113228, A113229, A117156, A117158, A176730, A201692, A201693, A231166.

Row m = 1 of A327722.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, 0), j=`if`(t<0, -t, 1)..u)+
      add(b(u+j-1, o-j, `if`(t=0, j, -j)), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 07 2013

A[x_]:=Integrate[AiryAi[ -t],{t,0,x}]; B[x_]:=Integrate[AiryBi[ -t],{t,0,x}];
c=-3^(2/3)*Gamma[2/3]/2; d=-3^(1/6)*Gamma[2/3]/2;
a[n_]:=SeriesCoefficient[1/(c*A[x]+d*B[x]+1),{x,0,n}]*n!; Table[a[n],{n,0,10}] (* fixed by Vaclav Kotesovec, Aug 23 2014 *)
(* constant d: *) 1/x/.FindRoot[3^(2/3)*Gamma[2/3]/2 * Integrate[AiryAi[-t],{t,0,x}] + 3^(1/6)*Gamma[2/3]/2 * Integrate[AiryBi[-t],{t,0,x}]==1,{x,1},WorkingPrecision->50] (* Vaclav Kotesovec, Aug 23 2014 *)

a(n) ~ c * d^n * n!, where d = 0.952891423325053197208702817349165942637814..., c = 1.169657787464830219717093446929792145316... . - Vaclav Kotesovec, Aug 23 2014
From Petros Hadjicostas, Nov 01 2019: (Start)
E.g.f.: 1/W(z), where W(z) := 1 + Sum_{n >= 0} (-1)^(n+1)* z^(3*n+1)/(b(n)*(3*n+1)) with b(n) = A176730(n) = (3*n)!/(3^n*(1/3)_n). (Here (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol, or rising factorial, which is denoted by (x)^n in some papers and books.) The function W(z) satisfies the o.d.e. W'''(z) + z*W'(z) = 0 with W(0) = 1, W'(0) = -1, and W''(0) = 0. [See Theorem 4.3 (Case 1243 with u = 0) in Elizalde and Noy (2003).]
a(n) = Sum_{m = 0..floor((n-1)/3)} (-3)^m * (1/3)_m * binomial(n, 3*m+1) * a(n-3*m-1) for n >= 1 with a(0) = 1. (End)

A014402 Numbers found in denominators of expansion of Airy function Ai(x).

Original entry on oeis.org

1, 1, 6, 12, 180, 504, 12960, 45360, 1710720, 7076160, 359251200, 1698278400, 109930867200, 580811212800, 46170964224000, 268334780313600, 25486372251648000, 161000868188160000, 17891433320656896000, 121716656350248960000, 15565546988971499520000, 113196490405731532800000

Offset: 0

N. J. A. Sloane

nonn

Although the description is technically correct, this sequence is unsatisfactory because there are gaps in the series.

A014402 arises via Vandermonde determinants as in A203433; see the Mathematica section. - Clark Kimberling, Jan 02 2012

			Mathematica gives the series as 1/(3^(2/3)*Gamma(2/3)) - x/(3^(1/3)*Gamma(1/3)) + x^3/(6*3^(2/3)*Gamma(2/3)) - x^4/(12*3^(1/3)*Gamma(1/3)) + x^6/(180*3^(2/3)*Gamma(2/3)) - x^7/(504*3^(1/3)*Gamma(1/3)) + x^9/(12960*3^(2/3)*Gamma(2/3)) - ...

G. C. Greubel, Table of n, a(n) for n = 0..420
NIST's Digital Library of Mathematical Functions, Airy and Related Functions (Maclaurin Series) by Frank W. J. Olver.

Cf. A014403, A060507, A176730, A176731, A203433, A203434.

A014402:= func< n | n eq 0 select 1 else (&*[n-j+Floor(n/2)-Floor(j/2): j in [0..n-1]]) >;
[A014402(n): n in [0..25]]; // G. C. Greubel, Sep 20 2023

Series[ AiryAi[ x ], {x, 0, 30} ]
a[ n_] := If[ n<0, 0, (n + Quotient[ n, 2])! / Product[ 3 k + 1 + Mod[n, 2], {k, 0, Quotient[ n, 2] - 1}]]; (* Michael Somos, Oct 14 2011 *)
(* Next, A014402 generated in via Vandermonde determinants based on A007494 *)
f[j_]:= j + Floor[(j+1)/2]; z = 20;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]             (* A203433 *)
Table[v[n+1]/v[n], {n,z}]      (* this sequence *)
Table[v[n]/d[n], {n,z}]        (* A203434 *)
(* Clark Kimberling, Jan 02 2012 *)

{a(n) = if( n<0, 0, (n\2 + n)! / prod( k=0, n\2 -1, n%2 + 3*k + 1))}; /* Michael Somos, Oct 14 2011 */

def A014402(n): return product(n-j+(n//2)-(j//2) for j in range(n))
[A014402(n) for n in range(31)] # G. C. Greubel, Sep 20 2023

a(2*n) = A176730(n). a(2*n + 1) = A176731(n). - Michael Somos, Oct 14 2011

A329070 Array read by ascending antidiagonals: T(n, k) = (kn)!/(k^n(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x(x + 1)...*(x + n - 1) is the Pochhammer symbol.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 8, 6, 1, 1, 48, 180, 24, 1, 1, 384, 12960, 8064, 120, 1, 1, 3840, 1710720, 10644480, 604800, 720, 1, 1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1, 1, 645120, 109930867200, 244635697152000, 2303884477440000, 70355755008000, 10897286400, 40320, 1

Offset: 0

Petros Hadjicostas, Nov 03 2019

For information about the function W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)) (mentioned in the Formula section below), see Theorem 3.2 in Elizalde and Noy (2003) with u = 0 and m and a in the theorem equal to our m + 1. See also the documentation of array A327722.
By using the ratio test and the Stirling approximation to the gamma function, we may show that the radius of convergence of the power series for W_m(z) is infinity (for each m >= 0). Thus, the function W_m(z) (as defined by the above power series) is entire.
If we define S(m,s) = T(n-s, s+1) for m >= 0 and 0 <= s <= m, we get the triangular array that appears in the Example section below.

			Array T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
  1,  1,     1,        1,           1,              1,  ...
  1,  2,     6,       24,         120,            720,  ...
  1,  8,   180,     8064,      604800,       68428800, ...
  1, 48, 12960, 10644480, 19813248000, 70355755008000, ...
  ...
Triangular array S(m,s) = T(m-s, s+1) (with rows m >= 0 and columns s >= 0):
  1;
  1,     1;
  1,     2,         1;
  1,     8,         6,           1;
  1,    48,       180,          24,           1;
  1,   384,     12960,        8064,         120,        1;
  1,  3840,   1710720,    10644480,      604800,      720,    1;
  1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1;
  ...

Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116).
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Wikipedia, Falling and rising factorials.

Rows include A000012 (n = 0), A000142 (n = 1), A060593 (n = 2).
Columns include A000012 (k = 1), A000165 (k = 2), A176730 (k = 3).
Ratios T(n+1,k)/(k!*T(n,k)) include A000012 (k = 1), A000027 (k = 2), A000326 (k = 3), A100157 (k = 4), A234043 (k = 5).
Cf. A111004, A117226, A202213, A327722.

A := (n, k) -> `if`(k=0, 1, (GAMMA(1/k)*GAMMA(k*n+1))/(GAMMA(n+1/k)*k^n)):
seq(seq(A(n-k-1, k), k=1..n-1), n=0..10); # Peter Luschny, Nov 04 2019

T(0,k) = 1, T(1,k) = k!, and T(2,k) = (2*k)!/(k + 1) for k >= 1.
T(n,1) = 1, T(n,2) = (2*n)!!, and T(n,3) is related to the Airy functions (see the documentation of A176730).
T(n+1,k) = (k-1)! * binomial(k*(n+1), k-1) * T(n,k) for n >= 0 and k >= 1.
T(n+1,k)/(k! * T(n,k)) = Cat(n+1, k), where Cat(d, k) = binomial(k*d, k)/(k * (d - 1) + 1) is a Fuss-Catalan number; see Theorem 1.2 in Schuetz and Whieldon (2014).
If F(k,z) = Sum_{n >= 0} z^(k*n)/T(n,k), then F(k,z) satisfies the o.d.e. F^(k-1)(k,z) - z*F(k,z) = 0.
If W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)), then 1/W_m(z) is the e.g.f. of row m of A327722(m,n), which counts permutations of [n] that avoid the consecutive pattern 12...(m+1)(m+3)(m+2) (or equivalently, the consecutive pattern (m+3)(m+2)...(3)(1)(2)).
The function W_m(z) satisfies the o.d.e. W_m^(m+2)(z) + z*W_m'(z) = 0 with W_m(0) = 1, W_m'(0) = -1, and W_m^(s)(0) = 0 for s = 2..(m + 1).

A176731 Denominators of coefficients of a series, called g, related to Airy functions.

Original entry on oeis.org

1, 12, 504, 45360, 7076160, 1698278400, 580811212800, 268334780313600, 161000868188160000, 121716656350248960000, 113196490405731532800000, 127006462235230779801600000, 169172607697327398695731200000, 263909268007830741965340672000000, 476620138022142319989405253632000000

Offset: 0

Wolfdieter Lang, Jul 14 2010

The numerators are always 1.
f(z) := Sum_{n>=0} (1/b(n)) * z^(3*n) with b(n) := A176730(n) and g(z) := Sum_{n>=0} (1/a(n)) * z^(3*n+1) build the two independent Airy functions Ai(z) = c(1)*f(z) - c(2)*g(z) and Bi(z) = sqrt(3) * (c(1)*f(z) + c(2)*g(z)) with c(1) := 1/(3^(2/3) * Gamma(2/3)), approximately 0.35502805388781723926, and c(2) := 1/(3^(1/3) * Gamma(1/3)), approximately 0.25881940379280679840.
If y := Sum_{n >= 0} x^(3*n+1)/a(n), then y'' = x*y. - Michael Somos, Jul 12 2019

			Rational g-coefficients: [1, 1/12, 1/504, 1/45360, 1/7076160, 1/1698278400, 1/580811212800, 1/268334780313600, ...].

G. C. Greubel, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 , 10.4.2 - 5. [alternative scanned copy].
Wolfdieter Lang, The first 20 terms of the f(z) and g(z) functions.
NIST's Digital Library of Mathematical Functions, Airy and Related Functions (Maclaurin Series) by Frank W. J. Olver.

Cf. A176730.

a := proc (n) option remember; if n = 0 then 1 else 3*n*(3*n+1)*a(n-1) end if; end proc: seq(a(n), n = 0..20); # Peter Bala, Dec 17 2021

a[ n_] := If[ n < 0, 0, -1 / (3^(1/3) Gamma[ 1/3] SeriesCoefficient[ AiryAi[ x], {x, 0, 3 n + 1}])]; (* Michael Somos, Oct 14 2011 *)
a[ n_] := If[ n < 0, 0, (3 n + 1)! / Product[ k, {k, 2, 3 n + 1, 3}]]; (* Michael Somos, Oct 14 2011 *)

{a(n) = if( n<0, 0, (3*n + 1)! / prod( k=0, n-1, 3*k + 2))}; /* Michael Somos, Oct 14 2011 */

a(n) = denominator((3^n) * risefac(2/3, n)/(3*n + 1)!) with the rising factorials risefac(k, n) := Product_{j=0..(n-1)} (k+j) and risefac(k, 0) = 1.
From Peter Bala, Dec 17 2021: (Start)
a(n) = 3*n*(3*n + 1)*a(n-1) with a(0) = 1.
a(n) = (3*n + 2)!/(n!*3^n)*Sum_{k = 0..n} (-1)^k*binomial(n,k)/(3*k + 2).
a(n) = (1/2)*(3*n + 2)!/(n!*3^n)*hypergeom([-n, 2/3], [5/3], 1).
a(n) = (2*Pi*sqrt(3))/9 *(1/3^(n+1))*Gamma(3*n+4)/( (n+1)*Gamma(1/3)* Gamma(n + 5/3) ). (End)
a(n) = (9^n*n!*(n + 1/3)!)/(1/3)!. - Peter Luschny, Dec 20 2021

A385253 Decimal expansion of the function AiryAi(x) at x=1.

Original entry on oeis.org

1, 3, 5, 2, 9, 2, 4, 1, 6, 3, 1, 2, 8, 8, 1, 4, 1, 5, 5, 2, 4, 1, 4, 7, 4, 2, 3, 5, 1, 5, 4, 6, 6, 3, 0, 6, 1, 7, 4, 9, 4, 4, 1, 4, 2, 9, 8, 8, 3, 3, 0, 7, 0, 6, 0, 0, 9, 1, 0, 2, 0, 5, 4, 7, 5, 7, 6, 3, 3, 5, 3, 4, 8, 0, 2, 2, 6, 5, 7, 2, 3, 6, 6, 3, 4, 8, 7, 1, 0, 9, 9, 0, 8, 7, 4, 8, 6, 8, 3, 2, 1, 3, 0, 5, 3

Offset: 0

Artur Jasinski, Jul 29 2025

Second derivative at x=1 has that same value.

			0.1352924163128814155241474235154663...

Cf. A014402, A176730, A284867, A284868, A269893.

AiryAi(1) ; evalf(%) ; # R. J. Mathar, Jul 31 2025

Mathematica
```
RealDigits[AiryAi[1], 10, 105][[1]]
```

airy(1)[1] \\ Michel Marcus, Jul 29 2025

Equals AiryAi''(1).

cp's OEIS Frontend

A117226 Number of permutations of [n] avoiding the consecutive pattern 1243.

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A014402 Numbers found in denominators of expansion of Airy function Ai(x).

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A329070 Array read by ascending antidiagonals: T(n, k) = (k*n)!/(k^n*(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol.

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A176731 Denominators of coefficients of a series, called g, related to Airy functions.

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A385253 Decimal expansion of the function AiryAi(x) at x=1.

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A329070 Array read by ascending antidiagonals: T(n, k) = (kn)!/(k^n(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x(x + 1)...*(x + n - 1) is the Pochhammer symbol.