A244589 -id:A244589 - cp's OEIS frontend

A263075 G.f. satisfies: [x^(n-1)] A(x)^(n^2) = n^n * (n-1)! for n>=1.

1, 1, 2, 31, 1150, 68713, 5914776, 692005074, 105932315154, 20617891510063, 4984425649932314, 1467604324373250545, 517561005098562714944, 215501019188749426210440, 104642607303457024105207408, 58625315029802441203026824094, 37541542090285460025870424920666

Offset: 0

Paul D. Hanna, Oct 08 2015

nonn

It appears that for k>0, a(k) is odd iff k = 2*A003714(n)+1 for n>=0, where A003714 is the fibbinary numbers (integers whose binary representation contains no consecutive ones); this is true for at least the first 531 terms. [See also A263190 and A171791.]

			G.f.: A(x) = 1 + x + 2*x^2 + 31*x^3 + 1150*x^4 + 68713*x^5 + 5914776*x^6 + 692005074*x^7 + 105932315154*x^8 +...
The coefficients in A(x)^(n^2) begin:
n=1: [1, 1, 2, 31, 1150, 68713, 5914776, 692005074, 105932315154, ...];
n=2: [1, 4, 14, 152, 5021, 289824, 24532494, 2841972672, 432284291486, ...];
n=3: [1, 9, 54, 507, 13356, 715635, 58722228, 6685822296, 1005887241243, ...];
n=4: [1, 16, 152, 1536, 31500, 1468016, 114260704, 12668897920, ...];
n=5: [1, 25, 350, 4275, 75000, 2840855, 202155100, 21547156900, ...];
n=6: [1, 36, 702, 10776, 184725, 5598720, 344795598, 34598389248, ...];
n=7: [1, 49, 1274, 24647, 456386, 11753973, 592950960, 54103596918, ...];
n=8: [1, 64, 2144, 51712, 1092016, 26366656, 1071635712, 84557168640, ...];
n=9: [1, 81, 3402, 100791, 2482650, 61309629, 2096140032, 135856780686, ...]; ...
where the terms along the main diagonal begin:
[1, 4, 54, 1536, 75000, 5598720, 592950960, 84557168640, ..., n^n*(n-1)!, ...].
Note that odd terms a(n) occur at positions n starting with:
[0, 1, 3, 5, 9, 11, 17, 19, 21, 33, 35, 37, 41, 43, 65, 67, 69, 73, 75, 81, 83, 85, 129, 131, 133, 137, 139, 145, 147, 149, 161, 163, 165, 169, 171, 257, ...],
which seems to equal A118113, the even Fibbinary numbers + 1, with an initial zero included.

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A171791, A263190, A003714, A118113, A244589.

{a(n) = local(A=[1,1]); for(i=1,n+1, A=concat(A,0); m=#A; A[m] = ( m^m*(m-1)! - Vec(Ser(A)^(m^2))[m] )/m^2 );A[n+1]}
for(n=0,20,print1(a(n),", "))

a(n) ~ exp(1-exp(-1)) * n! * n^(n-1). - Vaclav Kotesovec, Oct 20 2020

A244577 G.f. A(x) satisfies the property that the sum of the coefficients of x^k, k=0..n, in A(x)^n equals (n+1)!.

Original entry on oeis.org

1, 1, 2, 14, 196, 4652, 166168, 8232296, 535974416, 44186331248, 4489336764064, 550549455440096, 80153857492836928, 13665883723351362752, 2697370187692768024448, 610301579538939633684608, 156933087218604923576672512, 45515622704384079509089136384, 14789652457653705738777659937280

Offset: 0

Paul D. Hanna, Jun 30 2014

nonn

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 196*x^4/4! + 4652*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS.
If we form an array of coefficients of x^k/k! in A(x)^n, n>=0, like so:
A^0: [1],0,  0,   0,     0,      0,       0,         0,           0, ...;
A^1: [1, 1], 2,  14,   196,   4652,  166168,   8232296,   535974416, ...;
A^2: [1, 2,  6], 40,   528,  11824,  403840,  19373792,  1232259840, ...;
A^3: [1, 3, 12,  84], 1068,  22716,  741456,  34375200,  2132407248, ...;
A^4: [1, 4, 20, 152,  1912], 39008, 1218496,  54513152,  3292657664, ...;
A^5: [1, 5, 30, 250,  3180,  62980],1889080,  81499400,  4785873360, ...;
A^6: [1, 6, 42, 384,  5016,  97632, 2826288],117620256,  6706638336, ...;
A^7: [1, 7, 56, 560,  7588, 146804, 4127200, 165911312], 9177810320, ...;
A^8: [1, 8, 72, 784, 11088, 215296, 5918656, 230372480, 12358846848], ...; ...
then we can illustrate how the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals (n+1)!:
1! = 1;
2! = 1 + 1;
3! = 1 + 2 + 6/2!;
4! = 1 + 3 + 12/2! + 84/3!;
5! = 1 + 4 + 20/2! + 152/3! + 1912/4!;
6! = 1 + 5 + 30/2! + 250/3! + 3180/4! + 62980/5!; ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..160

Cf. A244589, A232606, A232607, A232683, A232687.

/* By Definition (slow): */
{a(n)=if(n==0, 1, n!*((n+1)! - sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j/j!)^n + x*O(x^k), k)))/n)}
for(n=0, 20, print1(a(n), ", "))

/* Faster, using series reversion: */
{a(n)=local(B=sum(k=0, n+1, (k+1)!*x^k)+x^3*O(x^n), G=1+x*O(x^n));
for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); n!*polcoeff(x/serreverse(x*G), n)}
for(n=0, 30, print1(a(n), ", "))

Given g.f. A(x), Sum_{k=0..n} [x^k] A(x)^n = (n+1)!.

a(n) ~ exp(-1) * (n!)^2. - Vaclav Kotesovec, Jul 03 2014

cp's OEIS Frontend

A263075 G.f. satisfies: [x^(n-1)] A(x)^(n^2) = n^n * (n-1)! for n>=1.

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A244577 G.f. A(x) satisfies the property that the sum of the coefficients of x^k, k=0..n, in A(x)^n equals (n+1)!.

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Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula