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 1-8 of 8 results.

A102693 a(n) is the number of digraphs (not allowing loops) with vertices 1,2,...,n that have a unique Eulerian tour (up to cyclic shift).

Original entry on oeis.org

1, 5, 42, 504, 7920, 154440, 3603600, 98017920, 3047466240, 106661318400, 4151586700800, 177925144320000, 8326896754176000, 422590010274432000, 23118159385601280000, 1356265350621941760000, 84945040381058457600000, 5657339689378493276160000
Offset: 2

Views

Author

Richard Stanley, Feb 04 2005

Keywords

Comments

It appears that a(n) can be obtained from the permanent of (2,3,4,...,n+2) as in A203470. - Clark Kimberling, Jan 02 2012

Examples

			a(3) = 5. There are two such digraphs that are triangles and three that consist of two 2-cycles with a common vertex.

References

  • R. P. Stanley, unpublished work.

Links

Crossrefs

Cf. A000108, A065866, A203470, A262034.

Programs

  • Maple
    with(combstruct):ZL:=[T,{T=Union(Z,Prod(Epsilon,Z,T), Prod(T,Z,Epsilon),Prod(T,T,Z))},labeled]: seq(count(ZL,size=i)/(2*i),i=2..18); # Zerinvary Lajos, Dec 16 2007
# alternative Maple program:
a:= proc(n) option remember; `if`(n<3, (n-1)*n/2,
       2*(n-1)*(2*n-1)*a(n-1)/(n+1))
    end:
seq(a(n), n=2..20);  # Alois P. Heinz, Nov 03 2017
  • Mathematica
    a[n_] := a[n] = If[n<3, n(n-1)/2, 2(n-1)(2n-1) a[n-1]/(n+1)];
Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Jun 10 2018, after Alois P. Heinz *)

Formula

a(n) = (1/2)*A065866(n-1).
a(n) = C_n(n-1)!/2 = (n+2)(n+3)...(2n-1), where C_n denotes a Catalan number.
E.g.f.: Integral_{x} 2/(1+sqrt(1-4*x))^2 dx. - Alois P. Heinz, Sep 09 2015
a(n) = RisingFactorial(4 + n, n) assuming offset 0. - Peter Luschny, Mar 22 2022.
Sum_{n>=2} 1/a(n) = (25*exp(1/4)*sqrt(Pi)*erf(1/2) - 10)/8, where erf is the error function. - Amiram Eldar, Dec 04 2022

A370058 a(n) = 4*(4*n+3)!/(3*n+4)!.

Original entry on oeis.org

1, 4, 44, 840, 23256, 850080, 38750400, 2120489280, 135566323200, 9922550077440, 818544054182400, 75160674504115200, 7604312776752384000, 840608992488545280000, 100812386907863414784000, 13037431708092153922560000, 1808675231786149165350912000
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2024

Keywords

Crossrefs

Cf. A365340, A370056, A370057.
Cf. A065866, A370055.
Cf. A000142, A002293.

Programs

  • PARI
    a(n) = 4*(4*n+3)!/(3*n+4)!;

Formula

E.g.f.: exp( Sum_{k>=1} binomial(4*k,k) * x^k/k ).
a(n) = A000142(n) * A002293(n+1).
D-finite with recurrence 3*(3*n+2)*(3*n+4)*(n+1)*a(n) -8*n*(4*n+1)*(2*n+1)*(4*n+3)*a(n-1)=0. - R. J. Mathar, Feb 22 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^(3/4))^4.
a(n) = 4 * Sum_{k=0..n} (3*n+4)^(k-1) * |Stirling1(n,k)|. (End)
E.g.f.: (1/x) * Series_Reversion( x/(1 + x)^4 ). - Seiichi Manyama, Feb 06 2025

A370055 a(n) = 3*(3*n+2)!/(2*n+3)!.

Original entry on oeis.org

1, 3, 24, 330, 6552, 171360, 5581440, 218045520, 9945936000, 519177859200, 30535045632000, 1998518736614400, 144098325316915200, 11350405033583616000, 969837188805041356800, 89351761457237190912000, 8830056426362263572480000, 931769828125956695715840000
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2024

Keywords

Crossrefs

Cf. A001763, A370054.
Cf. A065866, A370058.
Cf. A076151.
Cf. A000142, A001764.

Programs

  • PARI
    a(n) = 3*(3*n+2)!/(2*n+3)!;

Formula

E.g.f.: exp( Sum_{k>=1} binomial(3*k,k) * x^k/k ).
a(n) = 3*A076151(n+1) for n > 0.
a(n) = A000142(n)*A001764(n+1). - Alois P. Heinz, Feb 08 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^(2/3))^3.
a(n) = 3 * Sum_{k=0..n} (2*n+3)^(k-1) * |Stirling1(n,k)|. (End)
E.g.f.: (1/x) * Series_Reversion( x/(1 + x)^3 ). - Seiichi Manyama, Feb 06 2025

A256116 Number T(n,k) of length 2n k-ary words, either empty or beginning with the first letter of the alphabet and using each letter at least once, that can be built by repeatedly inserting doublets into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 9, 10, 0, 1, 34, 112, 84, 0, 1, 125, 930, 1800, 1008, 0, 1, 461, 7018, 26400, 35640, 15840, 0, 1, 1715, 51142, 334152, 816816, 840840, 308880, 0, 1, 6434, 368464, 3944220, 15550080, 27824160, 23063040, 7207200
Offset: 0

Views

Author

Alois P. Heinz, Mar 15 2015

Keywords

Examples

			T(3,2) = 9: aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba.
T(3,3) = 10: aabbcc, aabccb, aacbbc, aaccbb, abbacc, abbcca, abccba, acbbca, accabb, accbba.
T(4,2) = 34: aaaaaabb, aaaaabba, aaaabaab, aaaabbaa, aaaabbbb, aaabaaba, aaabbaaa, aaabbabb, aaabbbba, aabaaaab, aabaabaa, aabaabbb, aababbab, aabbaaaa, aabbaabb, aabbabba, aabbbaab, aabbbbaa, aabbbbbb, abaaaaba, abaabaaa, abaababb, abaabbba, ababbaba, abbaaaaa, abbaaabb, abbaabba, abbabaab, abbabbaa, abbabbbb, abbbaaba, abbbbaaa, abbbbabb, abbbbbba.
T(4,4) = 84: aabbccdd, aabbcddc, aabbdccd, aabbddcc, aabccbdd, aabccddb, aabcddcb, aabdccdb, aabddbcc, aabddccb, aacbbcdd, aacbbddc, aacbddbc, aaccbbdd, aaccbddb, aaccdbbd, aaccddbb, aacdbbdc, aacddbbc, aacddcbb, aadbbccd, aadbbdcc, aadbccbd, aadcbbcd, aadccbbd, aadccdbb, aaddbbcc, aaddbccb, aaddcbbc, aaddccbb, abbaccdd, abbacddc, abbadccd, abbaddcc, abbccadd, abbccdda, abbcddca, abbdccda, abbddacc, abbddcca, abccbadd, abccbdda, abccddba, abcddcba, abdccdba, abddbacc, abddbcca, abddccba, acbbcadd, acbbcdda, acbbddca, acbddbca, accabbdd, accabddb, accadbbd, accaddbb, accbbadd, accbbdda, accbddba, accdbbda, accddabb, accddbba, acdbbdca, acddbbca, acddcabb, acddcbba, adbbccda, adbbdacc, adbbdcca, adbccbda, adcbbcda, adccbbda, adccdabb, adccdbba, addabbcc, addabccb, addacbbc, addaccbb, addbbacc, addbbcca, addbccba, addcbbca, addccabb, addccbba.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    2;
  0, 1,    9,    10;
  0, 1,   34,   112,     84;
  0, 1,  125,   930,   1800,   1008;
  0, 1,  461,  7018,  26400,  35640,  15840;
  0, 1, 1715, 51142, 334152, 816816, 840840, 308880;

Links

Crossrefs

Columns k=0-2 give: A000007, A057427, A010763(n-1) for n>0.
Main diagonal gives A065866(n-1) (for n>0).
Row sums give A294603.
Cf. A183135, A256117.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
      add(binomial(2*n, j) *(n-j) *(k-1)^j, j=0..n-1))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)/
    `if`(k=0, 1, k):
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    Unprotect[Power]; 0^0 = 1; A[n_, k_] := A[n, k] = If[n==0, 1, k/n*Sum[ Binomial[2*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}]];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]/If[k==0, 1, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

Formula

T(n,k) = (Sum_{i=0..k} (-1)^i * C(k,i) * A183135(n,k-i)) / A028310(k).
T(n,k) = (k-1)! * A256117(n,k) for k > 0.

A380646 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x)/(1 + x)^2 ).

Original entry on oeis.org

1, 4, 46, 932, 27568, 1080432, 52916176, 3115326496, 214470890496, 16914853191680, 1504252282653184, 148956086481767424, 16256865070022066176, 1938988214539948730368, 250943399365390735104000, 35026523834624205803491328, 5245178283068781060488298496, 838841884254236846183525646336
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2025

Keywords

Links

Crossrefs

Cf. A088690, A380647, A380648.
Cf. A065866, A377829.
Cf. A097629, A380828.
Cf. A377892.

Programs

  • Mathematica
    nmax=18; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-2*x]/(1 + x)^2 ,{x,0,nmax}]],x]Range[0,nmax-1]! (* Stefano Spezia, Feb 06 2025 *)
  • PARI
    a(n) = 2*n!*sum(k=0, n, (2*n+2)^(k-1)*binomial(2*n+2, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * exp(2 * x * A(x)).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377892.
a(n) = 2 * n! * Sum_{k=0..n} (2*n+2)^(k-1) * binomial(2*n+2,n-k)/k!.

A110371 a(n)=[(n+1)(n+2)(n+3)...(2n)]/(1+2+3+...+n).

Original entry on oeis.org

2, 4, 20, 168, 2016, 31680, 617760, 14414400, 392071680, 12189864960, 426645273600, 16606346803200, 711700577280000, 33307587016704000, 1690360041097728000, 92472637542405120000, 5425061402487767040000, 339780161524233830400000, 22629358757513973104640000
Offset: 1

Views

Author

Amarnath Murthy, Jul 24 2005

Keywords

Comments

a(n)=2(n-1)!*Catalan(n). a(n)=2*A065866(n-1). - Emeric Deutsch, Aug 05 2005

Examples

			a(4) = 5*6*7*8/10 = 168.
a(5) = 10*9*8*7*6/(5+4+3+2+1) = 2016.

Links

  • Ron Knott, 2016, January 2016.

Crossrefs

Cf. A065866.

Programs

  • Maple
    seq(2*(2*n)!/(n+1)!/n,n=1..20); # Emeric Deutsch, Aug 05 2005
  • Mathematica
    Table[(Times@@Range[n+1,2n])/((n(n+1))/2),{n,20}] (* or *) Table[ 2(n-1)! CatalanNumber[n],{n,20}] (* Harvey P. Dale, Jul 15 2016 *)
  • PARI
    A110371(n)=binomial(2*n,n-1)/n*(n-1)!*2 \\ M. F. Hasler, Jan 31 2016

Formula

{2*(2n)!}/{(n+1)!*n}

Extensions

More terms from Emeric Deutsch, Aug 05 2005
Edited by M. F. Hasler, Jan 31 2016

A370326 E.g.f.: exp(Sum_{k>=1} binomial(2*k,k) * x^k).

Original entry on oeis.org

1, 2, 16, 200, 3376, 71552, 1822144, 54131072, 1836436480, 70016026112, 2962490758144, 137711245058048, 6974788150104064, 382232015239454720, 22531888624878813184, 1421482338801856053248, 95553266255536369893376, 6817598649041309962600448, 514534725049116493981941760
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 15 2024

Keywords

Crossrefs

Cf. A000984, A065866, A213507, A251568, A370327.

Programs

  • Mathematica
    CoefficientList[Series[Exp[Sum[Binomial[2*k,k]*x^k, {k, 1, 20}]], {x, 0, 20}], x] * Range[0, 20]!
CoefficientList[Series[Exp[1/Sqrt[1 - 4*x] - 1], {x, 0, 20}], x] * Range[0, 20]!

Formula

E.g.f.: exp(1/sqrt(1 - 4*x) - 1).
a(n) ~ exp(3*n^(1/3)/2^(2/3) - n - 1) * 2^(2*n + 1/6) * n^(n - 1/3) / sqrt(3).

A375867 E.g.f. satisfies A(x) = exp( 2 * (exp(x*A(x)^(1/2)) - 1) ).

Original entry on oeis.org

1, 2, 10, 82, 950, 14324, 266994, 5940218, 153797742, 4545958914, 151125136298, 5583189029004, 226989660492422, 10073099346726602, 484570780412539874, 25120235800280494530, 1396186059626363259038, 82828021612821756140124
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Crossrefs

Cf. A001861, A030019, A065866.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, (n+2)^(k-1)*stirling(n, k, 2));

Formula

a(n) = 2 * Sum_{k=0..n} (n+2)^(k-1) * Stirling2(n,k).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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