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.

Previous Showing 11-17 of 17 results.

A326437 E.g.f.: exp(-5) * Sum_{n>=0} (2*exp(n*x) + 3)^n / n!.

Original entry on oeis.org

1, 12, 298, 11154, 568004, 37059182, 2978383982, 286712714932, 32370944416718, 4216616929161674, 625354679867770896, 104450484419292872298, 19469192354728354857686, 4018460441266469063161936, 912287005016859245973405858, 226476227666270561445555706042, 61164205107875867322971316940164
Offset: 0

Views

Author

Paul D. Hanna, Jul 11 2019

Keywords

Comments

More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 3/2, r = 2.

Examples

			E.g.f.: A(x) = 1 + 12*x + 298*x^2/2! + 11154*x^3/3! + 568004*x^4/4! + 37059182*x^5/5! + 2978383982*x^6/6! + 286712714932*x^7/7! + 32370944416718*x^8/8! + 4216616929161674*x^9/9! + ...
such that
A(x) = exp(-5) * (1 + (2*exp(x) + 3) + (2*exp(2*x) + 3)^2/2! + (2*exp(3*x) + 3)^3/3! + (2*exp(4*x) + 3)^4/4! + (2*exp(5*x) + 3)^5/5! + (2*exp(6*x) + 3)^6/6! + ...)
also
A(x) = exp(-5) * (exp(3) + 2*exp(x)*exp(3*exp(x)) + 2^2*exp(4*x)*exp(3*exp(2*x))/2! + 2^3*exp(9*x)*exp(3*exp(3*x))/3! + 2^4*exp(16*x)*exp(3*exp(4*x))/4! + 2^5*exp(25*x)*exp(3*exp(5*x))/5! + 2^6*exp(36*x)*exp(3*exp(6*x))/6! + ...).

Crossrefs

Cf. A326600, A020557, A326433, A326434, A326435, A326436.

Programs

  • PARI
    /* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-5) * sum(n=0, 500, (2*exp(n*x +O(x^31)) + 3)^n/n! ) )))

Formula

E.g.f.: exp(-5) * Sum_{n>=0} (2*exp(n*x) + 3)^n / n!.
E.g.f.: exp(-5) * Sum_{n>=0} 2^n * exp(n^2*x) * exp( 3*exp(n*x) ) / n!.

A099977 Bisection of Bell numbers, A000110.

Original entry on oeis.org

1, 5, 52, 877, 21147, 678570, 27644437, 1382958545, 82864869804, 5832742205057, 474869816156751, 44152005855084346, 4638590332229999353, 545717047936059989389, 71339801938860275191172
Offset: 0

Views

Author

N. J. A. Sloane, Nov 19 2004

Keywords

Crossrefs

Cf. A000110, A020557.

Programs

  • Maple
    G:=series(exp(exp(x)-1),x=0,50): seq((2*n-1)!*coeff(G,x^(2*n-1)),n=1..18);
  • Python
    from itertools import accumulate, islice
def A099977_gen(): # generator of terms
    yield 1
    blist, b = (1,2), 1
    while True:
        for _ in range(2):
            blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b
A099977_list = list(islice(A099977_gen(),30)) # Chai Wah Wu, Jun 22 2022

Formula

E.g.f.: exp(-1)*Sum_{n>=0} n*exp(n^2*x)/n!. - Vladeta Jovovic, Aug 24 2006
a(n) = exp(-1) * Sum_{k>=0} k^(2*n+1)/k!. - Ilya Gutkovskiy, Jun 13 2019

Extensions

More terms from Emeric Deutsch, Dec 07 2004

A070906 Every third Bell number A000110.

Original entry on oeis.org

1, 5, 203, 21147, 4213597, 1382958545, 682076806159, 474869816156751, 445958869294805289, 545717047936059989389, 846749014511809332450147, 1629595892846007606764728147, 3819714729894818339975525681317
Offset: 0

Views

Author

Benoit Cloitre, May 19 2002

Keywords

Crossrefs

Cf. A000110, A020557.

Programs

  • Mathematica
    Table[ BellB[3*n], {n, 0, 12}] (* Jean-François Alcover, Dec 13 2012 *)
BellB[3*Range[0,15]] (* Harvey P. Dale, Apr 19 2020 *)
  • PARI
    for(n=0,50,print1(round(sum(i=0,1000,i^(3*n)/(i)!)/exp(1)),","))
  • Python
    from itertools import accumulate, islice
def A070906_gen(): # generator of terms
    yield 1
    blist, b = (1,), 1
    while True:
        for _ in range(3):
            blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b
A070906_list = list(islice(A070906_gen(),30)) # Chai Wah Wu, Jun 22 2022
  • Sage
    [bell_number(3*n) for n in range(0, 13)] # Zerinvary Lajos, May 14 2009

Formula

a(n) = Bell(3*n) = A000110(3*n). - Vladeta Jovovic, Feb 02 2003
a(n) = exp(-1)*Sum_{k>=0} k^(3n)/k!.
E.g.f.: exp(x*(d_z)^3)*(exp(exp(z)-1)) |_{z=0}, with the derivative operator d_z := d/dz. Adapted from eqs. (14) and (15) of the 1999 C. M. Bender reference given in A000110.
E.g.f.: exp(-1)*Sum_{n>=0} exp(n^3*x)/n!. - Vladeta Jovovic, Aug 24 2006

A108462 Number of factorizations of (n,n) into pairs (i,j) with i,j >= 1, not both 1.

Original entry on oeis.org

1, 2, 2, 9, 2, 15, 2, 31, 9, 15, 2, 92, 2, 15, 15, 109, 2, 92, 2, 92, 15, 15, 2, 444, 9, 15, 31, 92, 2, 203, 2, 339, 15, 15, 15, 712, 2, 15, 15, 444, 2, 203, 2, 92, 92, 15, 2, 1903, 9, 92, 15, 92, 2, 444, 15, 444, 15, 15, 2, 1663, 2, 15, 92, 1043, 15, 203, 2, 92, 15, 203, 2
Offset: 1

Views

Author

Christian G. Bower, Jun 03 2005

Keywords

Comments

The rule of building products is (a,b)*(x,y) = (a*x,b*y).
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

Examples

			From _Alois P. Heinz_ and _Antti Karttunen_, Nov 24 2017: (Start)
a(4) = 9 because for pair (4,4) there are nine factorizations:
  (4,4)
  (1,4)*(4,1)
  (1,2)*(4,2)
  (2,1)*(2,4)
  (2,2)*(2,2)
  (1,2)*(2,1)*(2,2)
  (1,4)*(2,1)*(2,1)
  (4,1)*(1,2)*(1,2)
  (1,2)*(1,2)*(2,1)*(2,1)
(End)
a(pq) = 15 for primes p<>q: (pq,pq); (p,1)(q,pq); (p,1)(q,1)(1,pq); (p,1)(q,1)(1,p)(1,q); (p,1)(q,q)(1,p); (p,1)(q,p)(1,q); (p,q)(q,p); (p,q)(q,1)(1,p); (p,p)(q,q) ; (p,p)(q,1)(1,q); (p,pq)(q,1); (pq,1)(1,pq); (pq,1)(1,p)(1,q); (pq,q)(1,p); (pq,p)(1,q). - _R. J. Mathar_, Nov 30 2017

Links

Crossrefs

Main diagonal of A108461.
Cf. A002774, A020557, A108463.

Programs

  • PARI
    a(n) = if(n==1, return(1)); my(b, c, r, x, y, v=List([]), w=List([[n]])); while(#w>r, c++; for(k=r+1, r=#w, y=w[k]; if(!isprime(x=y[c]), fordiv(x, d, if(d!=1&&d!=x, listput(w, concat([y[1..c-1], d, x/d]))))))); for(i=1, #w, x=w[i]; r=#x; for(j=1, #w, y=w[j]; for(k=0, 2^r-1, b=concat(b=binary(k), vector(r-#b)); if(#y>=t=vecsum(b), c=0; listput(v, vecsort(vector(r+#y-t, m, if(m>r, [1, y[m-r+t]], if(b[m], [x[m], y[c++]], [x[m], 1]))))))))); #Set(v); \\ Jinyuan Wang, Jan 17 2022

Formula

a(A025487(n)) = A108463(n).
a(p^k) = A002774(k).
a(A002110(n)) = A020557(n).
a(n) = A108461(n,n).

A308646 a(n) = exp(1) * Sum_{k>=0} (-1)^k*k^(2*n)/k!.

Original entry on oeis.org

1, 0, 1, -9, 50, 413, -17731, 110176, 9938669, -278475061, -9816860358, 725503033401, 15823587507881, -2848115497132448, -38795579403211671, 17235101634895315375, 153440975825762815938, -156894403296377741177371, -1454252568471818731501051, 2071137586315785548669378432
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 13 2019

Keywords

Links

Crossrefs

Cf. A000587, A020557, A308647.

Programs

  • Maple
    seq(BellB(2*n,-1),n=0..30); # Robert Israel, Jun 08 2020
  • Mathematica
    Table[Exp[1] Sum[(-1)^k k^(2 n)/k!, {k, 0, Infinity}], {n, 0, 19}]
Table[BellB[2 n, -1], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..2*n} (-1)^k*Stirling2(2*n,k).
a(n) = A000587(2*n).

A340822 a(n) = exp(-1) * Sum_{k>=0} (k*(k + n))^n / k!.

Original entry on oeis.org

1, 3, 43, 1211, 54812, 3572775, 313493737, 35368945463, 4962511954307, 844198388785291, 170675800745636572, 40352181663578992883, 11008690527354504977193, 3426969405868832970281647, 1205708016597226199323015459, 475502109963529414669658708847
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2021

Keywords

Crossrefs

Cf. A000110, A020557, A094577, A134980, A334242, A340823.

Programs

  • Mathematica
    Table[Exp[-1] Sum[(k (k + n))^n/k!, {k, 0, Infinity}], {n, 0, 15}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[2 n - k] n^k, {k, 0, n}], {n, 1, 15}]]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(2*n-k) * n^k.

A340823 a(n) = exp(-1) * Sum_{k>=0} (k*(k - n))^n / k!.

Original entry on oeis.org

1, 1, 3, 5, 124, -2075, 91993, -4709903, 312334595, -25531783799, 2524083665172, -296260739274275, 40667620527027177, -6446882734412545043, 1167717545574222779643, -239452569059443831797303, 55146244227862697483251020, -14163492441645773105212592623
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2021

Keywords

Links

Crossrefs

Cf. A000110, A020556, A020557, A290219, A334243, A340822.

Programs

  • Magma
    A340823:= func< n | (&+[(-n)^j*Binomial(n,j)*Bell(2*n-j): j in [0..n]]) >;
[A340823(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024
  • Mathematica
    Table[Exp[-1] Sum[(k (k - n))^n/k!, {k, 0, Infinity}], {n, 0, 17}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[2 n - k] (-n)^k, {k, 0, n}], {n, 1, 17}]]
  • SageMath
    def A340823(n): return sum( binomial(n,k)*bell_number(2*n-k)*(-n)^k for k in range(n+1))
[A340823(n) for n in range(31)] # G. C. Greubel, Jun 12 2024

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(2*n-k) * (-n)^k.
Previous Showing 11-17 of 17 results.

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

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