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.

A138912 Inverse binomial transform of A138911.

Original entry on oeis.org

1, 1, 2, 9, 28, 145, 726, 4249, 27000, 186561, 1387450, 11034001, 93295236, 834720913, 7870851366, 77943848025, 808159323376, 8749874083585, 98687599614450, 1157060036429857, 14075106913366140, 177337182574590801, 2310567819180310558, 31087556340427928617
Offset: 0

Views

Author

Paul D. Hanna, Apr 05 2008

Keywords

Comments

The n-th term of the n-th inverse binomial transform of A138911 is 1 for n>=0.

Crossrefs

Cf. A138911 (binomial transform).

Programs

  • PARI
    {a(n)=local(A=[1]);for(k=1,n,A=concat(A,0); A[k+1]=1-polcoeff(subst(Ser(A),x,x/(1+(k-1)*x+x*O(x^k)))/(1+(k-1)*x),k));A[n+1]}

Formula

O.g.f. satisfies: [x^n] A( x/(1+(n-1)*x) )/(1+(n-1)*x) = 1 for n>=0.
E.g.f. satisfies: [x^n] A(x)*exp(-(n-1)*x) = 1/n! for n>=0.

A138737 The n-th term of the n-th inverse binomial transform of this sequence equals (n+1)^(n-1) for n>=0.

Original entry on oeis.org

1, 2, 7, 52, 541, 7446, 127939, 2641192, 63746169, 1762380010, 54938528191, 1906911695580, 72949449568021, 3049813346508670, 138352912908850683, 6769028553912294736, 355311287187804226033, 19918243846821103623378
Offset: 0

Views

Author

Paul D. Hanna, Apr 05 2008

Keywords

Comments

Related to LambertW(-x)/(-x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

Examples

			If the successive inverse binomial transforms are placed in a table,
then we see that the diagonal consists of terms (n+1)^(n-1):
n=0:[(1),2,7,52,541,7446,127939,2641192,63746169,1762380010,...];
n=1:[1,(1),4,36,368,5200,90432,1884736,45817088,1273874688,...];
n=2:[1, 0,(3),26,245,3684,64087,1349214,33003945,922386824,...];
n=3:[1,-1, 4,(16),160,2688,45184,970240,23814144,668975104,...];
n=4:[1,-2,7, 0,(125),2002,31203,705268,17177273,486100710,...];
n=5:[1,-3,12,-28, 176,(1296),21184,524352,12305664,354510080,...];
n=6:[1,-4,19,-74,373, 0,(16807),395866,8645673,260994628,...];
n=7:[1,-5,28,-144,800,-2816, 24192,(262144),5980160,195969024,...];
n=8:[1,-6,39,-244,1565,-8562,56419, 0,(4782969),149083874,...];
n=9:[1,-7,52,-380,2800,-19248,136768,-638912, 6966528,(100000000),..];
n=10:[1,-8,67,-558,4661,-37604,302679,-2112938,17204009, 0,...].
Notice the occurrence of zeros in the secondary diagonal = A138734.

Links

Crossrefs

Cf. A138736 (inverse binomial transform), A138734; variants: A138909, A138911.

Programs

  • PARI
    {a(n)=local(A=[1]);for(k=1,n,A=concat(A,0); A[k+1]=(k+1)^(k-1)-Vec(subst(Ser(A),x,x/(1+k*x+x*O(x^k)))/(1+k*x))[k+1]);A[n+1]}

Formula

O.g.f. satisfies: [x^n] A( x/(1+n*x) )/(1+n*x) = (n+1)^(n-1) for n>=0.
E.g.f. satisfies: [x^n] A(x)*exp(-n*x) = (n+1)^(n-1)/n! for n>=0.
a(n) ~ (1 + LambertW(exp(-1)))^(3/2) * n^(n-1) / (exp(n-1) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Oct 30 2017

A138909 Expansion of e.g.f.: (1+x)/(1-x*exp(x)).

Original entry on oeis.org

1, 2, 6, 33, 232, 2045, 21636, 267043, 3766848, 59776137, 1053986500, 20442543671, 432537117552, 9914571191005, 244742376434388, 6473030199939675, 182614314495736576, 5473825396372806545, 173728330742517310980
Offset: 0

Views

Author

Paul D. Hanna, Apr 05 2008, Apr 06 2008

Keywords

Comments

The n-th term of the n-th inverse binomial transform of this sequence equals n! for n>=0.

Examples

			If the successive inverse binomial transforms are placed in a table,
then we see that the diagonal consists of the factorials:
n=0:[(1),2,6,33,232,2045,21636,267043,3766848,59776137,1053986500,...];
n=1:[1,(1),3,20,129,1164,12265,151458,2136337,33901640,597761361,...];
n=2:[1,0,(2),13,64,693,6856,86175,1210896,19228825,339012304,...];
n=3:[1,-1,3,(6),25,482,3429,50908,678465,10937430,192150469,...];
n=4:[1,-2,6,-7,(24),381,844,36291,341728,6433865,107801436,...];
n=5:[1,-3,11,-32,97,(120),-839,37158,55953,4638052,54573025,...];
n=6:[1,-4,18,-75,304,-811,(720),40783,-262608,5542425,6069736,...];
n=7:[1,-5,27,-142,729,-3282,11941,(5040),-497279,9166130,...];
n=8:[1,-6,38,-239,1480,-8643,45844,-178557,(40320),12301705,...];
n=9:[1,-7,51,-372,2689,-18844,125289,-741974,3354513,(362880),...].

Crossrefs

Cf. A006153.
Cf. A138910 (inverse binomial transform); variants: A138911, A138737.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(1+x)/(1-x*Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 09 2018 *)
  • PARI
    {a(n)=local(A=[1]);for(k=1,n,A=concat(A,0); A[k+1]=k!-polcoeff(subst(Ser(A),x,x/(1+k*x+x*O(x^k)))/(1+k*x),k));A[n+1]}
  • PARI
    {a(n)=n!+sum(k=0,n-1,k!*binomial(n,k)*n*k^(n-k-1))}

Formula

E.g.f. (1+x)/(1-x*exp(x)) - Olivier Gérard, Sep 15 2016
O.g.f. satisfies: [x^n] A( x/(1+n*x) )/(1+n*x) = n! for n>=0.
E.g.f. satisfies: [x^n] A(x)*exp(-n*x) = 1 for n>=0.
a(n) = n! + Sum_{k=0..n-1} k!*C(n,k)*n*k^(n-k-1) for n>1 with a(0)=1.
Equivalent to the sum above by properties of the binomial triangle:
a(n) = A006153(n)+n*A006153(n-1).
a(n) = n! ( Sum_{k=0..n-1} ((n-1-k)^k + (n-k)^k)/k!) for n>1 with a(0)=1.
a(n) ~ n! / LambertW(1)^n. - Vaclav Kotesovec, Oct 30 2017

Extensions

Name change and e.g.f. by Olivier Gérard, Sep 15 2016

A242598 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x-k)^k for 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 2, 5, 1, 2, 14, 10, 1, 2, 30, 58, 17, 1, 2, 55, 258, 167, 26, 1, 2, 91, 978, 1247, 386, 37, 1, 2, 140, 3330, 7862, 4306, 772, 50, 1, 2, 204, 10498, 44150, 40146, 11972, 1394, 65, 1, 2, 285, 31234, 227858, 330450, 153722, 28610, 2333, 82, 1, 2, 385, 88834, 1102658, 2480850, 1728722, 482210, 61133, 3682, 101, 1
Offset: 0

Views

Author

Derek Orr, Oct 15 2014

Keywords

Comments

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-0)^0 + A_1*(x-1)^1 + A_2*(x-2)^2 + ... + A_n*(x-n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

Examples

			1;
2,  1;
2,  5,      1;
2,  14,    10,       1;
2,  30,    58,      17,       1;
2,  55,   258,     167,      26,       1;
2,  91,   978,    1247,     386,      37,      1;
2, 140,  3330,    7862,    4306,     772,     50,     1;
2, 204, 10498,   44150,   40146,   11972,   1394,    65,    1;
2, 285, 31234,  227858,  330450,  153722,  28610,  2333,   82,   1;
2, 385, 88834, 1102658, 2480850, 1728722, 482210, 61133, 3682, 101, 1

Crossrefs

Cf. A000330, A002522, A138911, A248826.

Programs

  • PARI
    for(n=0,20,for(k=0,n,if(!k,if(n,print1(2,", "));if(!n,print1(1,", ")));if(k,print1(sum(i=1,n,(k^(i-k)*i*binomial(i,k)))/k,", "))))

Formula

T(n,1) = n*(2*n+1)*(n+1)/6 for n > 0.
T(n,n-1) = n^2 + 1 for n > 0.
Rows sum to SUM{k=0..n} A138911(k).
Showing 1-4 of 4 results.

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