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.

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208
Offset: 0

Views

Author

Johannes W. Meijer, Jul 27 2016

Keywords

Comments

The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
To preserve the identity IML[MNL[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).

Examples

			Some a(n) formulas, see A036039:
  a(0) = 1
  a(1) = 1*x(1)
  a(2) = 1*x(2) + 1*x(1)^2
  a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
  a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
  a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Links

  • M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
  • M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
  • N. J. A. Sloane, Transforms.
  • Eric W. Weisstein MathWorld, Exponential Transform.

Crossrefs

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

Programs

  • Maple
    nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(add(b(n)*x^n/n, n = 1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n = 0..nmax); # End second MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.
  • Mathematica
    b[n_] := (2*n - 1)!!^2;
a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)

Formula

a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].

A244432 E.g.f.: exp( Sum_{n>=1} Pell(n)*x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 1, 3, 17, 137, 1437, 18547, 284221, 5042289, 101635289, 2294115299, 57323597289, 1570795420537, 46836600355573, 1509632295204243, 52303597825637333, 1938434314587648353, 76521195859545355569, 3205495988651927796931, 142018837513142207290561
Offset: 0

Views

Author

Paul D. Hanna, Jun 27 2014

Keywords

Comments

a(n) == 0 (mod 7^k) for n >= 7*k, for k>=1 (conjecture).

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3/3! + 137*x^4/4! + 1437*x^5/5! + ...
where
log(A(x)) = x + 2*x^2/2 + 5*x^3/3 + 12*x^4/4 + 29*x^5/5 + 70*x^6/6 + 169*x^7/7 + 408*x^8/8 + 985*x^9/9 + ... + A000129(n)*x^n/n + ...

Links

Crossrefs

Cf. A000129, A244430.

Programs

  • PARI
    {a(n)=n!*polcoeff(exp(intformal(1/(1-2*x-x^2 +x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: ( (1 + x/S) / (1 - S*x) )^(sqrt(2)/4) where S = sqrt(2) + 1.
E.g.f.: exp( Integral 1/(1-2*x-x^2) dx ).
a(n) ~ n! * 2^(3*sqrt(2)/8) * n^(sqrt(2)/4-1) * (1+sqrt(2))^(n-1/(2*sqrt(2))) / GAMMA(1/(2*sqrt(2))). - Vaclav Kotesovec, Jun 28 2014
a(0) = a(1) = 1; a(n) = (2*n-1) * a(n-1) + (n-1) * (n-2) * a(n-2). - Ilya Gutkovskiy, Aug 13 2021
E.g.f.: exp((1/sqrt(2)) * arctanh(x*sqrt(2)/(1-x))). - Fabian Pereyra, Oct 11 2023
a(n) = n!*Sum_{k=0..n} binomial(n-1,k-1)*binomial(1/sqrt(8),k)*(1+sqrt(2))^(n-k)*(sqrt(8))^k. - Fabian Pereyra, Oct 19 2023

A244451 E.g.f.: exp( Sum_{n>=1} Fibonacci(2*n)*x^n/n ).

Original entry on oeis.org

1, 1, 4, 26, 236, 2756, 39376, 665464, 12986416, 287394416, 7112021696, 194607175136, 5834321568064, 190181750900416, 6697115871398656, 253362903806266496, 10248299242094541056, 441359565949128552704, 20163160035504969573376, 973917774772339989408256
Offset: 0

Views

Author

Paul D. Hanna, Jun 28 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 26*x^3/3! + 236*x^4/4! + 2756*x^5/5! +...
where
log(A(x)) = x + 3*x^2/2 + 8*x^3/3 + 21*x^4/4 + 55*x^5/5 + 144*x^6/6 + 377*x^7/7 + 987*x^8/8 +...+ A000045(2*n)*x^n/n +...

Links

Crossrefs

Cf. A244430, A000045.

Programs

  • PARI
    {a(n)=n!*polcoeff(exp(intformal(1/(1-3*x+x^2 +x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: ( (1 - x/Phi^2) / (1 - Phi^2*x) )^(sqrt(5)/5) where Phi = (sqrt(5)+1)/2.
E.g.f.: exp( Integral 1/(1-3*x+x^2) dx ).
a(n) ~ n! * 5^(1/(2*sqrt(5))) * n^(1/sqrt(5)-1) * ((1+sqrt(5))/2)^(2*n-2/sqrt(5)) / GAMMA(1/sqrt(5)). - Vaclav Kotesovec, Jun 28 2014

A291287 a(0)=a(1)=1, a(2)=3, thereafter a(n) = n*a(n-1)+(n-1)*(n-2)*a(n-2).

Original entry on oeis.org

1, 1, 3, 11, 62, 442, 3892, 40504, 487496, 6655688, 101656592, 1717234432, 31789038304, 639932442976, 13918144177088, 325239867277952, 8126648153635712, 216210586758515584, 6102238859442194176, 182102977877507458048, 5729025247479379569152
Offset: 0

Views

Author

N. J. A. Sloane, Aug 27 2017

Keywords

Links

Crossrefs

Cf. A244430, A291286.

Programs

  • Maple
    f:=proc(n) option remember;
if n <= 1 then 1 elif n=2 then 3 else
n*f(n-1)+(n-1)*(n-2)*f(n-2); fi; end;
[seq(f(n),n=0..20)];
  • Mathematica
    nxt[{n_,a_,b_}]:={n+1,b,b(n+1)+n(n-1)a}; Join[{1},NestList[nxt,{2,1,3},20][[;;,2]]] (* Harvey P. Dale, Jan 20 2024 *)

A274539 E.g.f.: exp(sum(bell(n)*z^n/n, n=1..infinity)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2079, 38629, 951187, 29979753, 1175837345, 56066617331, 3187704802281, 212628685506643, 16413606252207007, 1449425836362499605, 144977415195565990619, 16285937949513614300369, 2039447464767566886933057, 282862729890000953318773603
Offset: 0

Views

Author

Johannes W. Meijer, Jun 29 2016

Keywords

Comments

The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
Some transform pairs, see the formula section, are: x(n) = A000027(n) and a(n) = A000262(n); x(n) = A000045(n) and a(n) = A244430(n); x(n) = A000079(n) and a(n) = A000165(n); x(n) = A000108(n) and a(n) = A213507(n); x(n) = A000142(n) and a(n) = A158876(n); x(n) = A000203(n) and a(n) = A053529(n).

Crossrefs

Cf. A036039, A000110, A000165, A000262, A053529, A158876, A213507, A244430.

Programs

  • Maple
    a := proc(n): n!*P(n) end: P := proc(n): if n=0 then 1 else P(n):= expand((1/n)*(add(x(n-k) * P(k), k=0..n-1))) fi; end: with(combinat): x := proc(n): bell(n) end: seq(a(n), n=0..18);
  • Mathematica
    nmax = 20; CoefficientList[Series[E^(Sum[BellB[n]*z^n/n, {n, 1, nmax}]), {z, 0, nmax}], z] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 29 2016 *)

Formula

a(n) = n! * P(n), with P(n) = (1/n)*(sum(x(n-k) * P(k), k=0..n-1)), n >=1 and P(0) = 1, with x(n) = A000110(n), the Bell numbers.
E.g.f.: exp(sum(x(n)*z^n/n, n=1..infinity)) with x(n) = A000110(n).

A291286 a(0)=1, a(1)=2, thereafter a(n) = n*a(n-1)+(n-1)*(n-2)*a(n-2).

Original entry on oeis.org

1, 2, 4, 16, 88, 632, 5552, 57824, 695776, 9500128, 145097152, 2451080192, 45373649024, 913400022656, 19865889564928, 464227147597312, 11599471170191872, 308605525316616704, 8709955613991289856, 259922447412719218688, 8177253768239405504512
Offset: 0

Views

Author

N. J. A. Sloane, Aug 27 2017

Keywords

Crossrefs

A244430 and A291287 have the same recurrence.

Programs

  • Maple
    f:=proc(n) option remember;
if n <= 1 then n+1 else
n*f(n-1)+(n-1)*(n-2)*f(n-2); fi; end;
[seq(f(n),n=0..40)];
Showing 1-6 of 6 results.

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

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