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 1-6 of 6 results.

A080108 a(n) = Sum_{k=1..n} k^(n-k)*binomial(n-1,k-1).

Original entry on oeis.org

1, 2, 6, 23, 104, 537, 3100, 19693, 136064, 1013345, 8076644, 68486013, 614797936, 5818490641, 57846681092, 602259154853, 6548439927680, 74180742421185, 873588590481988, 10674437936521069, 135097459659312176
Offset: 1

Views

Author

Vladeta Jovovic, Mar 15 2003

Keywords

Comments

Row sums of triangle A154372. Example: a(3)=1+12+9+1=23. From A152818. - Paul Curtz, Jan 08 2009
Number of pointed set partitions of a pointed set k[1...k...n] with a prescribed point k. - Gus Wiseman, Sep 27 2015
With offset 0, a(n) is the number of partial functions (A000169) from [n]->[n] such that every element in the domain of definition is mapped to a fixed point. This implies a(n) is the number of idempotent partial functions Cf. A121337. - Geoffrey Critzer, Aug 07 2016

Examples

			G.f. = x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 537*x^6 + 3100*x^7 + 19693*x^8 + ...
The a(4) = 23 pointed set partitions of 1[1 2 3 4] are 1[1[1 2 3 4]], 1[1[1] 2[2 3 4]], 1[1[1] 3[2 3 4]], 1[1[1] 4[2 3 4]], 1[1[1 2] 3[3 4]], 1[1[1 2] 4[3 4]], 1[1[1 3] 2[2 4]], 1[1[1 3] 4[2 4]], 1[1[1 4] 2[2 3]], 1[1[1 4] 3[2 3]], 1[1[1 2 3] 4[4]], 1[1[1 2 4] 3[3]], 1[1[1 3 4] 2[2]], 1[1[1] 2[2] 3[3 4]], 1[1[1] 2[2] 4[3 4]], 1[1[1] 2[2 3] 4[4]], 1[1[1] 2[2 4] 3[3]], 1[1[1] 3[3] 4[2 4]], 1[1[1] 3[2 3] 4[4]], 1[1[1 2] 3[3] 4[4]], 1[1[1 3] 2[2] 4[4]], 1[1[1 4] 2[2] 3[3]], 1[1[1] 2[2] 3[3] 4[4]].

Links

Crossrefs

First column of array A098697.
Cf. A152818, A185298, A262671.

Programs

  • Magma
    [(1/n)*(&+[Binomial(n,k)*k^(n-k+1): k in [0..n]]): n in [1..30]]; // G. C. Greubel, Jan 22 2023
  • Mathematica
    Table[Sum[k^(n-k) Binomial[n-1,k-1],{k,n}],{n,30}] (* Harvey P. Dale, Aug 19 2012 *)
Table[SeriesCoefficient[Sum[x^k/(1-k*x)^k,{k,0,n}],{x,0,n}], {n,1,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
CoefficientList[Series[E^(x*(1+E^x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 06 2014 *)
  • PARI
    a(n)=sum(k=1,n, k^(n-k)*binomial(n-1,k-1)) \\ Anders Hellström, Sep 27 2015
  • SageMath
    def A080108(n): return (1/n)*sum(binomial(n,k)*k^(n-k+1) for k in range(n+1))
[A080108(n) for n in range(1,31)] # G. C. Greubel, Jan 22 2023

Formula

G.f.: Sum_{k>0} x^k/(1-k*x)^k.
E.g.f. (for offset 0): exp(x*(1+exp(x))). - Vladeta Jovovic, Aug 25 2003
a(n) = A185298(n)/n.

A361540 Expansion of e.g.f. A(x,y) satisfying A(x,y) = Sum_{n>=0} (A(x,y)^n + y)^n * x^n/n!, as a triangle read by rows.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 22, 39, 18, 1, 269, 604, 426, 92, 1, 4616, 12625, 12040, 4550, 520, 1, 102847, 332766, 401355, 218300, 50085, 3222, 1, 2824816, 10574725, 15456756, 11017895, 3867080, 577731, 21700, 1, 92355769, 393171416, 676130644, 597596216, 284455150, 69038984, 7022596, 157544, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 20 2023

Keywords

Comments

A202999(n) = Sum_{k=0..n} T(n,k).
A361053(n) = Sum_{k=0..n} T(n,k) * 2^k.
A361054(n) = Sum_{k=0..n} T(n,k) * 3^k.
A361055(n) = Sum_{k=0..n} T(n,k) * 4^k.
A361056(n) = Sum_{k=0..n} T(n,k) * 2^(n-k).
A361057(n) = Sum_{k=0..n} T(n,k) * 3^(n-k).
A203013(n) = Sum_{k=0..n} T(n,k) * 2^(n-k) * (-1)^k.
A155806(n) = T(n,0) for n >= 0; e.g.f. G(x) = Sum_{n>=0} G(x)^(n^2)*x^n/n!.
A361544(n) = T(n,1) for n >= 1.
A361549(n) = T(n,2) for n >= 2.
A185298(n) = T(n,n-1) for n >= 1; e.g.f. x*exp(x)*exp(x*exp(x)).
A361539(n) = T(n,n-2) for n >= 2.
A361688(n) = T(2*n,n) / binomial(2*n,n) for n >= 0.

Examples

			E.g.f. A(x,y) = 1 + (y + 1)*x + (y^2 + 4*y + 3)*x^2/2! + (y^3 + 18*y^2 + 39*y + 22)*x^3/3! + (y^4 + 92*y^3 + 426*y^2 + 604*y + 269)*x^4/4! + (y^5 + 520*y^4 + 4550*y^3 + 12040*y^2 + 12625*y + 4616)*x^5/5! + (y^6 + 3222*y^5 + 50085*y^4 + 218300*y^3 + 401355*y^2 + 332766*y + 102847)*x^6/6! + (y^7 + 21700*y^6 + 577731*y^5 + 3867080*y^4 + 11017895*y^3 + 15456756*y^2 + 10574725*y + 2824816)*x^7/7! + (y^8 + 157544*y^7 + 7022596*y^6 + 69038984*y^5 + 284455150*y^4 + 597596216*y^3 + 676130644*y^2 + 393171416*y + 92355769)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k in e.g.f. A(x,y) begins:
[1];
[1, 1];
[3, 4, 1];
[22, 39, 18, 1];
[269, 604, 426, 92, 1];
[4616, 12625, 12040, 4550, 520, 1];
[102847, 332766, 401355, 218300, 50085, 3222, 1];
[2824816, 10574725, 15456756, 11017895, 3867080, 577731, 21700, 1];
[92355769, 393171416, 676130644, 597596216, 284455150, 69038984, 7022596, 157544, 1];
[3506278528, 16744363569, 33151425840, 35028273756, 21134516256, 7193104758, 1262445744, 90148860, 1224576, 1]; ...
RELATED TABLE.
The elements of this triangle T(n,k) when divided by binomial(n,k) yields the related triangle:
[1];
[1, 1];
[3, 2, 1];
[22, 13, 6, 1];
[269, 151, 71, 23, 1];
[4616, 2525, 1204, 455, 104, 1];
[102847, 55461, 26757, 10915, 3339, 537, 1];
[2824816, 1510675, 736036, 314797, 110488, 27511, 3100, 1];
[92355769, 49146427, 24147523, 10671361, 4063645, 1232839, 250807, 19693, 1];
[3506278528, 1860484841, 920872940, 417003259, 167734256, 57088133, 15029116, 2504135, 136064, 1]; ...

Links

Crossrefs

Cf. A202999 (y=1), A361053 (y=2), A361054 (y=3), A361055 (y=4), A361056, A361057, A203013.
Cf. A155806 (T(n,0)), A361544 (T(n,1)), A361549 (T(n,2)), A185298 (T(n,n-1)), A361539 (T(n,n-2)), A361688 (T(2*n,n)/C(2*n,n)).

Programs

  • PARI
    /* E.g.f. A(x,y) = Sum_{n>=0} (A(x,y)^n + y)^n * x^n/n! */
{T(n,k) = my(A = 1); for(i=1,n, A = sum(m=0, n, (A^m + y +x*O(x^n))^m * x^m/m! )); n!*polcoeff(polcoeff(A, n,x),k,y)}
for(n=0, 12, for(k=0, n, print1(T(n,k), ", ")); print(" "))
  • PARI
    /* E.g.f. A(x,y) = Sum_{n>=0} A(x,y)^(n^2) * exp(y*x*A(x,y)^n) * x^n/n! */
{T(n,k) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(y*x*A^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(polcoeff(A, n,x),k,y)}
for(n=0, 12, for(k=0, n, print1(T(n,k), ", ")); print(" "))

Formula

E.g.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n! may be defined as follows.
(1) A(x,y) = Sum_{n>=0} (A(x,y)^n + y)^n * x^n/n!.
(2) A(x,y) = Sum_{n>=0} A(x,y)^(n^2) * exp(y*x*A(x,y)^n) * x^n/n!.

A207833 E.g.f.: T(T(x)), where T(x) is the e.g.f. for labeled rooted trees, A000169.

Original entry on oeis.org

1, 4, 30, 332, 4880, 89742, 1986124, 51471800, 1530489744, 51395228090, 1924687118684, 79553145323940, 3598161485778808, 176797212122233094, 9378715234039802340, 534259395682874552048, 32528761111972930621472, 2108146039402630977388530, 144899759883703796130871468, 10528261771566724089621962780
Offset: 1

Views

Author

N. J. A. Sloane, Feb 20 2012

Keywords

Comments

Exponential series reversal gives A185298 with alternating signs: 1, -4, 18, -92, 520, ... . - Vladimir Reshetnikov, Aug 04 2019

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 30*x^3/3! + 332*x^4/4! + 4880*x^5/5! +...
Euler's tree function T(x) satisfies: T(x/exp(x)) = x, and begins:
T(x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! +...+ A000169(n)*x^n/n! +...
where e.g.f. A(x) = T(T(x)).

Links

Crossrefs

Cf. A000169, A185298, A227278, A227176.

Programs

  • Mathematica
    nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[ 0,nn]!CoefficientList[ ComposeSeries[ Series[t,{x,0,nn}],Series[t,{x,0,nn}]],x] (* Geoffrey Critzer, Sep 16 2012 *)
Rest[CoefficientList[Series[-LambertW[LambertW[-x]], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Feb 24 2014 *)
  • PARI
    {a(n)=if(n==0||n==1, 1, n^(n-1)-sum(k=1, n-1, (-1)^(n-k)*binomial(n, k)*k^(n-k)*a(k)))} \\ Paul D. Hanna, Nov 21 2012

Formula

a(n) = 1/n * Sum_{k=1..n} C(n,k)*k^k*n^(n-k). [Vladimir Kruchinin, Sep 24 2012]
a(n) = n^(n-1) - Sum_{k=1..n-1} (-1)^(n-k) * C(n, k) * k^(n-k) * a(k) for n>1 with a(1)=1. - Paul D. Hanna, Nov 21 2012
E.g.f. A(x) satisfies: A(x) = Sum_{n>=1} n^(n-1)*T(x)^n/n!, by definition.
E.g.f. A(x) satisfies: A(x/exp(x)) = T(x) = Sum_{n>=1} n^(n-1)*x^n/n!. - Paul D. Hanna, Jul 04 2013
a(n) ~ n^(n-1) * exp(n*exp(-1)) / sqrt(1-exp(-1)). - Vaclav Kotesovec, Feb 24 2014

A262671 Number of pointed multiset partitions of normal pointed multisets of weight n.

Original entry on oeis.org

1, 6, 42, 335, 2956, 28468, 296540
Offset: 1

Views

Author

Gus Wiseman, Sep 26 2015

Keywords

Comments

A pointed multiset k[1...k...n] with point k is normal if its entries [1...k...n] span an initial interval of positive integers. Pointed multiset partitions are triangles (or compositions) in the multiorder of pointed multisets.

Examples

			The a(2) = 6 pointed multiset partitions are:
1[1[11]],1[1[1]1[1]],
1[1[12]],1[1[1]2[2]],
2[2[12]],2[1[1]2[2]].
The a(3) = 42 pointed multiset partitions are:
1[1[111]],1[1[1]1[11]],1[1[11]1[1]],1[1[1]1[1]1[1]],
1[1[122]],1[1[1]2[22]],1[1[12]2[2]],1[1[1]2[2]2[2]],
2[2[122]],2[1[1]2[22]],2[1[12]2[2]],2[2[2]2[12]],2[2[12]2[2]],2[1[1]2[2]2[2]],
1[1[112]],1[1[1]1[12]],1[1[1]2[12]],1[1[11]2[2]],1[1[12]1[1]],1[1[1]1[1]2[2]],
2[2[112]],2[1[1]2[12]],2[1[11]2[2]],2[1[1]1[1]2[2]],
1[1[123]],1[1[1]2[23]],1[1[1]3[23]],1[1[12]3[3]],1[1[13]2[2]],1[1[1]2[2]3[3]],
2[2[123]],2[1[1]2[23]],2[1[13]2[2]],2[2[2]3[13]],2[2[12]3[3]],2[1[1]2[2]3[3]],
3[3[123]],3[1[1]3[23]],3[1[12]3[3]],3[2[2]3[13]],3[2[12]3[3]],3[1[1]2[2]3[3]].

Links

Crossrefs

Cf. A185298, A080108, A276024.

Programs

  • Mathematica
    ReplaceListRepeated[forms_List, rerules_List] :=
Union[Flatten[
   FixedPointList[
    Function[preforms,
     Union[Flatten[ReplaceList[#, rerules] & /@ preforms, 1]]],
    forms], 1]]
pointedPartitions[JIX[r_, b_List?OrderedQ]] /; MemberQ[b, r] :=
  Cases[ReplaceListRepeated[{Z[Y[JIX[r, {r}]],
      Y @@ DeleteCases[b, r, 1, 1]]}, {Z[Y[sof___, JIX[w_, t_]],
        Y[for___, x_, aft___]] /; OrderedQ[{w, x}] :>
      Z[Y[sof, JIX[w, t], JIX[x, {x}]], Y[for, aft]],
     Z[Y[JIX[w_, t_], soa___], Y[for___, x_, aft___]] /;
       OrderedQ[{x, w}] :>
      Z[Y[JIX[x, {x}], JIX[w, t], soa], Y[for, aft]],
     Z[Y[sof___, JIX[w_, {tof__}]], Y[for___, x_, aft___]] :>
      Z[Y[sof, JIX[w, Sort[{tof, x}]]], Y[for, aft]],
     Z[Y[JIX[w_, {tof__}], soa___], Y[for___, x_, aft___]] :>
      Z[Y[JIX[w, Sort[{tof, x}]], soa], Y[for, aft]]}],
   Z[Y[pts__], Y[]] :> JIX[r, {pts}]];
allnormpms[n_Integer] :=
  Join @@ Function[s,
     JIX[#, Array[Count[s, y_ /; y <= #] + 1 &, n]] & /@
      Range[Length[s] + 1]] /@ Subsets[Range[n - 1] + 1];
Join @@ pointedPartitions /@ allnormpms[3] /.
JIX -> Apply(* to construct the example *)
Array[Plus @@ (Length[pointedPartitions[#]] & /@
     allnormpms[#]) &, 7](* to compute the sequence *)

A361544 a(n) = A361540(n,1) for n >= 1, a column of triangle A361540.

Original entry on oeis.org

1, 4, 39, 604, 12625, 332766, 10574725, 393171416, 16744363569, 803841993370, 42957812253301, 2529951235854516, 162852898603253209, 11378885054925777494, 858009440175419213445, 69471138931959493061296, 6013997809048628612191585, 554545575488282609142617778
Offset: 1

Views

Author

Paul D. Hanna, Mar 20 2023

Keywords

Comments

E.g.f. F(x,y) of triangle A361540 satisfies the following.
(1) F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n!.
(2) F(x,y) = Sum_{n>=0} F(x,y)^(n^2) * exp(y*x*F(x,y)^n) * x^n/n!.
The column next to this one in triangle A361540 has e.g.f. G(x) = Sum_{n>=0} G(x)^(n^2)*x^n/n!.

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 39*x^3/3! + 604*x^4/4! + 12625*x^5/5! + 332766*x^6/6! + 10574725*x^7/7! + 393171416*x^8/8! + 16744363569*x^9/9! + 803841993370*x^10/10! + ... + a(n)*x^n/n! + ...
a(n) is divisible by n, where a(n)/n begins
[1, 2, 13, 151, 2525, 55461, 1510675, 49146427, 1860484841, ...].

Links

Crossrefs

Cf. A361540, A185298.

Programs

  • PARI
    /* E.g.f. of triangle A361540 is F(x,y) = Sum_{n>=0} (F(x,y)^n + y)^n * x^n/n! */
{A361540(n,k) = my(F = 1); for(i=1,n, F = sum(m=0, n, (F^m + y +x*O(x^n))^m * x^m/m! )); n!*polcoeff(polcoeff(F, n,x),k,y)}
for(n=1, 20, print1(A361540(n,1), ", "))

A376551 This sequence satisfies: n = Sum_{k=0..n} ((-n)^(n - k)*binomial(n, k)*a(k)).

Original entry on oeis.org

0, 1, 6, 30, 164, 980, 6342, 44254, 331144, 2642472, 22379210, 200311034, 1887949164, 18676191196, 193352093326, 2089583250990, 23519349939728, 275137968890576, 3339075981451410, 41967997127203042, 545452423113576820, 7320310586184404676, 101314914535943061206, 1444341387745444125590, 21185535150823665972120, 319401932972290702809400
Offset: 0

Views

Author

Thomas Scheuerle, Nov 27 2024

Keywords

Crossrefs

Cf. A000248, A185298.

Programs

  • PARI
    a(max_n) = {my(x='x+O('x^(max_n+1))); concat([0], Vec(serlaplace(x*exp(x)*exp(x*exp(x))*(1+x))))}
  • PARI
    a(n) = n*sum(k=0, n, binomial(n, k)*(n-k)^k)

Formula

E.g.f.: x*exp(x)*exp(x*exp(x))*(x+1).
a(n) = A000248(n)*n.
a(n) = n*Sum_{k=0..n} (binomial(n, k)*(n - k)^k).
Showing 1-6 of 6 results.

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

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