cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A208426 Expansion of Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-3*x)^(3*n+1).

Original entry on oeis.org

1, 3, 15, 99, 711, 5373, 42099, 338355, 2771127, 23028813, 193610385, 1643215005, 14056350075, 121040308665, 1048212778635, 9122168556819, 79727173530327, 699443806767525, 6156776010386481, 54356715121718349, 481194980656865721, 4270165015550478003
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2012

Keywords

Comments

Compare g.f. to: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-2*x)^(3*n+1), which is a g.f. of the Franel numbers (A000172).
Diagonal of rational functions 1/(1 - x*y - y*z - x*z - 3*x*y*z), 1/(1 - x*y + y*z + x*z - 3*x*y*z). - Gheorghe Coserea, Jul 04 2018
Diagonal of the rational function 1/(1 - (x^2 + y^2 + z^2 + 3*x*y*z)). - Seiichi Manyama, Jul 05 2025

Examples

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 99*x^3 + 711*x^4 + 5373*x^5 + 42099*x^6 + ...
where
A(x) = 1/(1-3*x) + 6*x^2/(1-3*x)^4 + 90*x^4/(1-3*x)^7 + 1680*x^6/(1-3*x)^10 + 34650*x^8/(1-3*x)^13 + 756756*x^10/(1-3*x)^16 + ...

Links

Crossrefs

Cf. A000172, A002893, A069835, A208425, A294035.

Programs

  • Mathematica
    Table[3^n * HypergeometricPFQ[{1/2 - n/2, -n/2, 1 + n}, {1, 1}, 4/9], {n, 0, 25}] (* Vaclav Kotesovec, Oct 07 2020 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)/(1-3*x+x*O(x^n))^(3*m+1)),n)}
for(n=0,31,print1(a(n),", "))
  • PARI
    a(n) = sum(k=0, n\2, (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k)); \\ Gheorghe Coserea, Jul 04 2018

Formula

From Gheorghe Coserea, Jul 04 2018: (Start)
a(n) = Sum_{k=0..floor(n/2)} (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k).
G.f. y=A(x) satisfies: 0 = x*(3*x + 2)*(27*x^3 + 9*x - 1)*y'' + (243*x^4 + 216*x^3 + 27*x^2 + 36*x - 2)*y' + 3*(27*x^3 + 33*x^2 - 2*x + 2)*y.
(End)
From Vaclav Kotesovec, Oct 07 2020: (Start)
Recurrence: n^2*(3*n - 5)*a(n) = 3*(9*n^3 - 24*n^2 + 17*n - 4)*a(n-1) + 3*(3*n - 4)*a(n-2) + 27*(n-2)^2*(3*n - 2)*a(n-3).
a(n) ~ sqrt(2 + sqrt(5)*phi^(-1/3) + sqrt(5)*phi^(1/3)) * 3^n * (1 + phi^(-2/3) + phi^(2/3))^n / (2*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
(End)

A294036 a(n) = 4^n*hypergeom([-n/4, (1-n)/4, (2-n)/4, (3-n)/4], [1, 1, 1], 1).

Original entry on oeis.org

1, 4, 16, 64, 280, 1504, 9856, 70144, 498136, 3449440, 23506816, 160566784, 1115048896, 7905796864, 56994288640, 414928113664, 3034880623576, 22255957312864, 163667338903936, 1208070406612480, 8955840250934080, 66678657938510080
Offset: 0

Views

Author

Peter Luschny, Nov 02 2017

Keywords

Comments

Diagonal of rational function 1/(1 - (x^4 + y^4 + z^4 + t^4 + 4*x*y*z*t)). - Gheorghe Coserea, Aug 04 2018

Crossrefs

H(1, n, 1) = A000007(n), H(2, n, 1) = A000984(n), H(3, n, 1) = A006077(n), H(4, n, 1) = this seq., H(1, n, -1) = A000079(n), H(2, n, -1) = A098335(n), H(3, n, -1) = A294035(n), H(4, n, -1) = A294037(n).

Programs

  • Maple
    T := (m,n,x) -> m^n*hypergeom([seq((k-n)/m, k=0..m-1)], [seq(1, k=0..m-2)], x):
lprint(seq(simplify(T(4,n,1)), n=0..39));
  • Mathematica
    Table[4^n * HypergeometricPFQ[{-n/4, (1-n)/4, (2-n)/4, (3-n)/4}, {1, 1, 1}, 1], {n, 0, 30}] (* Vaclav Kotesovec, Nov 02 2017 *)

Formula

Let H(m, n, x) = m^n*hypergeom([(k-n)/m for k=0..m-1], [1 for k=0..m-2], x) then a(n) = H(4, n, 1).
a(n) ~ 2^(3*n + 2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Nov 02 2017

A294037 a(n) = 4^n*hypergeom([-n/4, (1-n)/4, (2-n)/4, (3-n)/4], [1, 1, 1], -1).

Original entry on oeis.org

1, 4, 16, 64, 232, 544, -1664, -37376, -362024, -2743712, -17780864, -98955776, -442825664, -1129423616, 5536033792, 118591811584, 1224814969816, 9905491019104, 68032143081856, 398051159254528, 1854461906222272, 4784426026102528
Offset: 0

Views

Author

Peter Luschny, Nov 02 2017

Keywords

Comments

Diagonal of rational function 1/(1 - (x^4 + y^4 + z^4 - t^4 + 4*x*y*z*t)). - Gheorghe Coserea, Aug 04 2018

Crossrefs

H(1, n, 1) = A000007(n), H(2, n, 1) = A000984(n), H(3, n, 1) = A006077(n), H(4, n, 1) = A294036(n), H(1, n, -1) = A000079(n), H(2, n, -1) = A098335(n), H(3, n, -1) = A294035(n), H(4, n, -1) = this seq..

Programs

  • Maple
    T := (m,n,x) -> m^n*hypergeom([seq((k-n)/m, k=0..m-1)], [seq(1, k=0..m-2)], x):
lprint(seq(simplify(T(4,n,-1)), n=0..39));
  • Mathematica
    Table[4^n * HypergeometricPFQ[{-n/4, (1-n)/4, (2-n)/4, (3-n)/4}, {1, 1, 1}, -1], {n, 0, 20}] (* Vaclav Kotesovec, Nov 02 2017 *)

Formula

Let H(m, n, x) = m^n*hypergeom([(k-n)/m for k=0..m-1], [1 for k=0..m-2], x) then a(n) = H(4, n, -1).

A349001 The number of Lyndon words of size n from an alphabet of 5 letters and 1st and 2nd letter of the alphabet with equal frequency in the words.

Original entry on oeis.org

1, 3, 4, 14, 46, 174, 656, 2640, 10790, 45340, 193600, 839820, 3686424, 16353924, 73187456, 330052646, 1498335650, 6841899606, 31404443032, 144814450188, 670552118244, 3116578216310, 14534401932712, 67992210407514, 318969964124256, 1500268062754830
Offset: 0

Views

Author

R. J. Mathar, Nov 05 2021

Keywords

Comments

Counts a subset of the Lyndon words in A001692. Here there is no requirement of how often the 3rd to 5th letter of the alphabet are in the admitted word, only on the frequency of the 1st and 2nd letter of the alphabet.
Let T(n,k,M) be the number of words of length n drawn from an alphabet of size M where the first k letters of the alphabet appear with the same frequency f in each word. Then T(n,k,M) = Sum_{f=0..floor(n/k)} (M-k)^(n-f*k) * Product_{i=0..k-1} binomial(n-i*f,f) and T(n,2,5) = A026375(n), T(n,3,6) = A294035(n), T(n,2,6) = A081671(n). Removing the words with cycles by the inclusion-exclusion principle by a Mobius Transform gives words of length n of that type without cycles and division through n the Lyndon words of that type. - R. J. Mathar, Nov 07 2021

Examples

			Examples for the alphabet {0,1,2,3,4}:
a(0)=1 counts (), the empty word.
a(3)=14 counts (021) (031) (041) (012) (013) (223) (233) (243) (014) (224) (234) (334) (244) (344), words of length 3 where the letters 0 and the 1 occur both either not or once.
a(4)=46 counts (0011) (0221) (0321) (0421) (0231) (0331) (0431) (0241) (0341) (0441) (0212) (0312) (0412) (0122) (0132) (0142) (0213) (0313) (0413) (0123) (2223) (0133) (2233) (2333) (2433) (0143) (2243) (2343) (2443) (0214) (0314) (0414) (0124) (2224) (2324) (0134) (2234) (2334) (3334) (2434) (0144) (2244) (2344) (3344) (2444) (3444).

Links

Crossrefs

Cf. A022553 (alphabet of 2 letters), A290277 (of 3 letters), A060165 (of 4 letters), A026375.

Programs

  • PARI
    a(n) = if(n>0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, binomial(d,k)*binomial(2*k,k)))/n, n==0) \\ Andrew Howroyd, Jan 14 2023

Formula

n*a(n) = Sum_{d|n} mu(d)*A026375(n/d) where mu = A008683.

Extensions

Terms a(16) and beyond from Andrew Howroyd, Jan 14 2023
Showing 1-4 of 4 results.

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