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.

A264902 Number T(n,k) of defective parking functions of length n and defect k; triangle T(n,k), n>=0, 0<=k<=max(0,n-1), read by rows.

Original entry on oeis.org

1, 1, 3, 1, 16, 10, 1, 125, 107, 23, 1, 1296, 1346, 436, 46, 1, 16807, 19917, 8402, 1442, 87, 1, 262144, 341986, 173860, 41070, 4320, 162, 1, 4782969, 6713975, 3924685, 1166083, 176843, 12357, 303, 1, 100000000, 148717762, 96920092, 34268902, 6768184, 710314, 34660, 574, 1
Offset: 0

Views

Author

Alois P. Heinz, Nov 28 2015

Keywords

Examples

			T(2,0) = 3: [1,1], [1,2], [2,1].
T(2,1) = 1: [2,2].
T(3,1) = 10: [1,3,3], [2,2,2], [2,2,3], [2,3,2], [2,3,3], [3,1,3], [3,2,2], [3,2,3], [3,3,1], [3,3,2].
T(3,2) = 1: [3,3,3].
Triangle T(n,k) begins:
0 :       1;
1 :       1;
2 :       3,       1;
3 :      16,      10,       1;
4 :     125,     107,      23,       1;
5 :    1296,    1346,     436,      46,      1;
6 :   16807,   19917,    8402,    1442,     87,     1;
7 :  262144,  341986,  173860,   41070,   4320,   162,   1;
8 : 4782969, 6713975, 3924685, 1166083, 176843, 12357, 303, 1;
    ...

Links

Crossrefs

Columns k=0-10 give: A000272(n+1), A140647, A291128, A291129, A291130, A291131, A291132, A291133, A291134, A291135, A291136.
Row sums give A000312.
T(2n,n) gives A264903.
Cf. A036276, A101334, A274279.

Programs

  • Maple
    S:= (n, k)-> `if`(k=0, n^n, add(binomial(n, i)*k*
            (k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k)):
T:= (n, k)-> S(n, k)-S(n, k+1):
seq(seq(T(n, k), k=0..max(0, n-1)), n=0..10);
  • Mathematica
    S[n_, k_] := If[k==0, n^n, Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}]]; T[n_, k_] := S[n, k]-S[n, k+1]; T[0, 0] = 1; Table[T[n, k], {n, 0, 10}, {k, 0, Max[0, n-1]}] // Flatten (* Jean-François Alcover, Feb 18 2017, translated from Maple *)

Formula

T(n,k) = S(n,k) - S(n,k+1) with S(n,0) = n^n, S(n,k) = Sum_{i=0..n-k} C(n,i) * k*(k+i)^(i-1) * (n-k-i)^(n-i) for k>0.
Sum_{k>0} k * T(n,k) = A036276(n-1) for n>0.
Sum_{k>0} T(n,k) = A101334(n).
Sum_{k>=0} (-1)^k * T(n,k) = A274279(n) for n>=1.

A238085 Expansion of e.g.f.: -LambertW(-sinh(x)).

Original entry on oeis.org

0, 1, 2, 10, 72, 716, 9088, 140344, 2554240, 53540368, 1270296064, 33653698464, 984753299456, 31542901202112, 1097763264864256, 41247391653500800, 1664188908529156096, 71759140177774010624, 3293251384307726942208, 160272893566770148403712
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 17 2014

Keywords

Links

Crossrefs

Cf. A219503, A214654, A274279, A277463, A277476.

Programs

  • Mathematica
    CoefficientList[Series[-LambertW[-Sinh[x]],{x,0,20}],x]*Range[0,20]!
  • PARI
    x='x+('x^30); concat([0], Vec(serlaplace(-lambertw(-sinh(x))))) \\ G. C. Greubel, Feb 19 2018

Formula

a(n) ~ (exp(-2)+1)^(1/4) * n^(n-1) / ((log(sqrt(1+exp(-2)) + exp(-1)) )^(n-1/2) * exp(n-1/2)).

A195136 a(n) = ((n+1)^(n-1) + (n-1)^(n-1))/2 for n>=1.

Original entry on oeis.org

1, 2, 10, 76, 776, 9966, 154400, 2803256, 58388608, 1372684090, 35958682112, 1038736032324, 32805006411776, 1124535087475814, 41584800431742976, 1650158470945337584, 69943137585151901696, 3153813559835569475058, 150745204037648268787712, 7613458147995669857352380, 405143549343202022103973888, 22657085569540734204315357022, 1328470689420203636727039918080, 81494507575933974604289943213096, 5220210773193749540624447754469376, 348542314841685116176787263033063466, 24216786265392720787141148530274467840, 1748280517106781152846793195054531026356, 130956723831431687431286364126682302906368, 10164786953127554557192799138093559445158870
Offset: 1

Views

Author

Paul D. Hanna, Sep 09 2011

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 76*x^4/4! + 776*x^5/5! + 9966*x^6/6! + 154400*x^7/7! + 2803256*x^8/8! + 58388608*x^9/9! + 1372684090*x^10/10! +...
such that A(x) = sinh(x*W(x))
where W(x) = LambertW(-x)/(-x) begins
W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! +...+ (n+1)^(n-1)*x^n/n! +...
and satisfies W(x) = exp(x*W(x)).
Also, A(x) = (W(x) - 1/W(x))/2 where
1/W(x) = 1 - x - x^2/2! - 4*x^3/3! - 27*x^4/4! - 256*x^5/5! - 3125*x^6/6! - 46656*x^7/7! - 823543*x^8/8! +...+ -(n-1)^(n-1)*x^n/n! +...

Crossrefs

Cf. A000272, A274278, A274279.

Programs

  • Mathematica
    Join[{1},Table[((n+1)^(n-1)+(n-1)^(n-1))/2,{n,2,30}]] (* Harvey P. Dale, Feb 06 2023 *)
  • PARI
    {a(n)=((n+1)^(n-1) + (n-1)^(n-1))/2}
for(n=1,30,print1(a(n),", "))
  • PARI
    {a(n)=sum(k=0,(n-1)\2,binomial(n-1,2*k)*n^(n-2*k-1))}
for(n=1,30,print1(a(n),", "))
  • PARI
    {a(n)=local(W=sum(m=0,n,(m+1)^(m-1)*x^m/m!)+x*O(x^n));n!*polcoeff(sinh(x*W),n)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f.: sinh(x*W(x)) = (W(x) - 1/W(x))/2 where W(x) = LambertW(-x)/(-x) = exp(x*W(x)) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
a(n) = Sum_{k=0..floor((n-1)/2)} C(n-1,2*k) * n^(n-2*k-1).

Extensions

Entry revised by Paul D. Hanna, Jun 19 2016
Corrected and extended by Harvey P. Dale, Feb 06 2023

A274278 a(n) = ((n+1)^(n-1) - (n-1)^(n-1))/2 for n>=1.

Original entry on oeis.org

1, 0, 1, 6, 49, 520, 6841, 107744, 1979713, 41611392, 985263601, 25958682112, 753424361713, 23888905963520, 821659980883561, 30472793606184960, 1212264580564478209, 51496393511442350080, 2326573297949232710881, 111398795962351731212288, 5635038492335356268228401, 300285949343202022103973888, 16814498551154751682934232601, 987042812055984079330393194496
Offset: 0

Views

Author

Paul D. Hanna, Jun 19 2016

Keywords

Examples

			E.g.f.: A(x) = 1 + x^2/2! + 6*x^3/3! + 49*x^4/4! + 520*x^5/5! + 6841*x^6/6! + 107744*x^7/7! + 1979713*x^8/8! + 41611392*x^9/9! + 985263601*x^10/10! +...
such that A(x) = cosh(x*W(x))
where W(x) = LambertW(-x)/(-x) begins
W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! +...+ (n+1)^(n-1)*x^n/n! +...
and satisfies W(x) = exp(x*W(x)).
Also, A(x) = (W(x) + 1/W(x))/2 where
1/W(x) = 1 - x - x^2/2! - 4*x^3/3! - 27*x^4/4! - 256*x^5/5! - 3125*x^6/6! - 46656*x^7/7! - 823543*x^8/8! +...+ -(n-1)^(n-1)*x^n/n! +...

Crossrefs

Cf. A000272, A195136, A274279.

Programs

  • PARI
    {a(n) = ((n+1)^(n-1) - (n-1)^(n-1))/2}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n) = sum(k=0,(n-1)\2, binomial(n-1,2*k+1) * n^(n-2*k-2))}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n) = my(W=sum(m=0,n, (m+1)^(m-1)*x^m/m!) +x*O(x^n)); n!*polcoeff(cosh(x*W),n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f.: cosh(x*W(x)) = (W(x) + 1/W(x))/2 where W(x) = LambertW(-x)/(-x) = exp(x*W(x)) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
a(n) = Sum_{k=0..floor((n-1)/2)} C(n-1,2*k+1) * n^(n-2*k-2).
Showing 1-4 of 4 results.

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