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 1-5 of 5 results.

A107668 Column 0 of triangle A107667.

Original entry on oeis.org

1, 4, 45, 816, 20225, 632700, 23836540, 1048592640, 52696514169, 2976295383100, 186548057815801, 12845016620629488, 963644465255618276, 78224633235142116240, 6830914919397129328500, 638477522900795994967040, 63599377775480137499907561, 6725771848938288950491594140
Offset: 0

Views

Author

Paul D. Hanna, Jun 07 2005

Keywords

Comments

Shift right of column 1 of triangle A107670, which is the matrix square of triangle A107667.
The o.g.f. A(x) = Sum_{m >= 0} a(m)*x^m is such that, for each integer n > 0, the coefficient of x^n in the expansion of exp(n^2*x)*(1 - x*A(x)) is equal to 0.
Given the o.g.f. A(x), the o.g.f. of A304322 equals 1/(1 - x*A(x)).
Also, a(n) is the number of 2-symbol Turing Machine state graphs in which n states are reached in canonical order. A canonical TM state graph lists for each state 1..n, and each of 2 symbols 0,1 in lexicographic order, a next state that is either the halt state, an already listed state, or the least unlisted state, as in the Haskell program below. Multiplied by 4^(2*n), this gives a much smaller number of TMs to be considered for the Busy Beaver function than given by A052200. - John Tromp, Oct 15 2024

Examples

			O.g.f.: A(x) = 1 + 4*x + 45*x^2 + 816*x^3 + 20225*x^4 + 632700*x^5 + 23836540*x^6 + 1048592640*x^7 + 52696514169*x^8 + 2976295383100*x^9 + ...
From _Petros Hadjicostas_, Mar 10 2021: (Start)
We illustrate the above formula for a(n) with the compositions of n + 1 for n = 2.
The compositions of n + 1 = 3 are 3, 1 + 2, 2 + 1, and 1 + 1 + 1.  Thus the above sum has four terms with (r = 1, s_1 = 3), (r = 2, s_1 = 1, s_2 = 2), (r = 2, s_1 = 2, s_2 = 1), and (r = 3, s_1 = s_2 = s_3 = 1).
The value of the denominator Product_{j=1..r} s_j! for these four terms is 6, 2, 2, and 1, respectively.
The value of the numerator Product_{j=1..r} (Sum_{i=1..j} s_i)^(2*s_j) for these four terms is 729, 81, 144, and 36.
Thus a(2) = 729/6 - 81/2 - 144/2 + 36/1 = 45. (End)

Links

Crossrefs

Cf. A107667, A107669, A107670, A052200.
Cf. A304322, A107675, A304394, A304395, A342202.

Programs

  • Haskell
    -- using program for A107667
a107668 = map head a where a = [[sum [a!!n!!i * a!!i!!(k+1) | i<-[k+1..n]] | k <- [0..n-1]] ++ [fromIntegral n+1] | n <- [0..]] -- John Tromp, Oct 21 2024
  • Haskell
    -- low memory version
a107668 n = (foldl' (\r i->sum r`seq`listArray(0,n)(0:[if i+1<2*j then 0 else r!j*(n+2-j)+r!(j-1)|j<-[1..n]])) (listArray(0,n)(0:repeat 1)) [1..2*n])!n -- John Tromp, Oct 15 2024
  • PARI
    {a(n)=local(A);if(n==0,n+1,A=(n+1)*x+x*O(x^n); for(k=0,n,A+=polcoeff(A,k)*x^k*(n+1-prod(i=0,k,1+(i-n-1)*x))); polcoeff(A,n))}
for(n=0,30, print1(a(n),", "))
  • PARI
    /* From formula: [x^n] exp( n^2*x ) * (1 - x*A(x)) = 0 */
{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m^2 +x^2*O(x^m)) * (1 - x*Ser(A)) )[m+1] ); A[n+1]}
for(n=0,25, print1( a(n),", ")) \\ Paul D. Hanna, May 12 2018
  • PARI
    /* From Recurrence: */
{a(n) = if(n==0,1, (n+1)^(2*n+2)/(n+1)! - sum(k=1,n, (n+1)^(2*k)/k! * a(n-k) ))}
for(n=0,25, print1( a(n),", ")) \\ Paul D. Hanna, May 12 2018

Formula

O.g.f. A(x) satisfies: [x^n] exp( n^2*x ) * (1 - x*A(x)) = 0 for n > 0. - Paul D. Hanna, May 12 2018
a(n) = (n+1)^2 * A107669(n).
a(n) = (n+1)^(2*n+2)/(n+1)! - Sum_{k=1..n} (n+1)^(2*k)/k! * a(n-k) for n > 0 with a(0) = 1. - Paul D. Hanna, May 12 2018
a(n) = A342202(2,n+1) = Sum_{r=1..(n+1)} (-1)^(r-1) * Sum_{s_1, ..., s_r} (1/(Product_{j=1..r} s_j!)) * Product_{j=1..r} (Sum_{i=1..j} s_i)^(2*s_j)), where the second sum is over lists (s_1, ..., s_r) of positive integers s_i such that Sum_{i=1..r} s_i = n+1. (Thus the second sum is over all ordered partitions (i.e., compositions) of n+1. See Michel Marcus's PARI program in A342202.) - Petros Hadjicostas, Mar 10 2021
a(n) ~ sqrt(1-c) * 2^(2*n + 3/2) * n^(n + 1/2) / (sqrt(Pi) * exp(n) * c^(n+1) * (2-c)^(n+1)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Oct 18 2024

A107675 Column 0 of triangle A107674.

Original entry on oeis.org

1, 24, 2268, 461056, 160977375, 85624508376, 64363893844726, 64928246784463872, 84623205378726331245, 138408056280920732755000, 277597038523589348539241112, 670011760601512512626484887040
Offset: 0

Views

Author

Paul D. Hanna, Jun 07 2005

Keywords

Comments

The o.g.f. A(x) = Sum_{m >= 0} a(m)*x^m is such that, for each integer n > 0, the coefficient of x^n in the expansion of exp(n^3*x)*(1 - x*A(x)) is equal to 0.
Given the o.g.f. A(x), the o.g.f. of A304323 equals 1/(1 - x*A(x)).

Examples

			O.g.f.: A(x) = 1 + 24*x + 2268*x^2 + 461056*x^3 + 160977375*x^4 + 85624508376*x^5 + 64363893844726*x^6 + 64928246784463872*x^7 + ...

Crossrefs

Cf. A107671, A107674, A107676.
Cf. A304323, A107668, A304394, A304395, A342202.

Programs

  • PARI
    {a(n)=local(P=matrix(n+1,n+1,r,c,if(r>=c,(r^3)^(r-c)/(r-c)!)), D=matrix(n+1,n+1,r,c,if(r==c,r)));(P^-1*D^2*P)[n+1,1]}
for(n=0,20, print1(a(n),", "))
  • PARI
    /* From formula: [x^n] exp( n^3*x ) * (1 - x*A(x)) = 0 */
{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m^3 +x^2*O(x^m)) * (1 - x*Ser(A)) )[m+1] ); A[n+1]}
for(n=0, 25, print1( a(n), ", ")) \\ Paul D. Hanna, May 12 2018
  • PARI
    /* From Recurrence: */
{a(n) = if(n==0,1, (n+1)^(3*n+3)/(n+1)! - sum(k=1,n, (n+1)^(3*k)/k! * a(n-k) ))}
for(n=0,25, print1( a(n),", ")) \\ Paul D. Hanna, May 12 2018

Formula

O.g.f. A(x) satisfies: [x^n] exp(n^3*x) * (1 - x*A(x)) = 0 for n > 0. - Paul D. Hanna, May 12 2018
a(n) = (n+1)^(3*n+3)/(n+1)! - Sum_{k=1..n} (n+1)^(3*k)/k! * a(n-k) for n > 0 with a(0) = 1. - Paul D. Hanna, May 12 2018
a(n) = A342202(3,n+1) = Sum_{r=1..(n+1)} (-1)^(r-1) * Sum_{s_1, ..., s_r} (1/(Product_{j=1..r} s_j!)) * Product_{j=1..r} (Sum_{i=1..j} s_i)^(3*s_j)), where the second sum is over lists (s_1, ..., s_r) of positive integers s_i such that Sum_{i=1..r} s_i = n+1. (Thus the second sum is over all compositions of n+1. See Michel Marcus's PARI program in A342202.) - Petros Hadjicostas, Mar 10 2021

A304394 O.g.f. A(x) satisfies: [x^n] exp(n^4 * x) * (1 - x*A(x)) = 0 for n > 0.

Original entry on oeis.org

1, 112, 76221, 152978176, 673315202500, 5508710472669120, 75300988091046198131, 1595530380622638283804672, 49561200934127182294698009969, 2161539625780059763174286300310000, 127884966535158110582342524738392563401, 9979510403062963314615799917574094659938048, 1003426348756281631241586585232930123009989117616
Offset: 0

Views

Author

Paul D. Hanna, May 12 2018

Keywords

Comments

INVERT transform of A304324.
The o.g.f. A(x) = Sum_{m >= 0} a(m)*x^m is such that, for each integer n > 0, the coefficient of x^n in the expansion of exp(n^4 * x) * (1 - x*A(x)) = 0 is equal to 0.

Examples

			O.g.f.: A(x) = 1 + 112*x + 76221*x^2 + 152978176*x^3 + 673315202500*x^4 + 5508710472669120*x^5 + 75300988091046198131*x^6 + ...

Crossrefs

Cf. A304324, A304396, A107668, A107675, A304395, A342202.

Programs

  • PARI
    /* From formula: [x^n] exp( n^4*x ) * (1 - x*A(x)) = 0 */
{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m^4 +x^2*O(x^m)) * (1 - x*Ser(A)) )[m+1] ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Formula

a(n) = (n+1)^(4*n+4)/(n+1)! - Sum_{k=1..n} (n+1)^(4*k)/k! * a(n-k) for n > 0 with a(0) = 1.
a(n) = A342202(4,n+1) = Sum_{r=1..(n+1)} (-1)^(r-1) * Sum_{s_1, ..., s_r} (1/(Product_{j=1..r} s_j!)) * Product_{j=1..r} (Sum_{i=1..j} s_i)^(4*s_j)), where the second sum is over lists (s_1, ..., s_r) of positive integers s_i such that Sum_{i=1..r} s_i = n+1. (Thus the second sum is over all compositions of n+1. See Michel Marcus's PARI program in A342202.) - Petros Hadjicostas, Mar 10 2021

A304395 O.g.f. A(x) satisfies: [x^n] exp( n^5 * x ) * (1 - x*A(x)) = 0 for n > 0.

Original entry on oeis.org

1, 480, 2245320, 43083161600, 2331513459843750, 287128730182879382976, 69929145078323834449039740, 30496052356323314014140611297280, 22113924320024426907851753695581691875, 25177421842925471123473548283955430812500000, 42994775028354266041451477298870703788676694998956, 106089234738948935762581435147478647028049918327743508480
Offset: 0

Views

Author

Paul D. Hanna, May 12 2018

Keywords

Comments

The o.g.f. A(x) = Sum_{m >= 0} a(m)*x^m is such that, for each integer n > 0, the coefficient of x^n in the expansion of exp(n^5*x) * (1 - x*A(x)) is equal to 0.

Examples

			O.g.f.: A(x) = 1 + 480*x + 2245320*x^2 + 43083161600*x^3 + 2331513459843750*x^4 + 287128730182879382976*x^5 + 69929145078323834449039740*x^6 + ...

Crossrefs

Cf. A304325, A304397, A107668, A107675, A304394.
INVERT transform of A304325.

Programs

  • PARI
    /* From formula: [x^n] exp( n^5*x ) * (1 - x*A(x)) = 0 */
{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m^5 +x^2*O(x^m)) * (1 - x*Ser(A)) )[m+1] ); A[n+1]}
for(n=0, 20, print1( a(n), ", "))

Formula

a(n) = (n+1)^(5*n+5)/(n+1)! - Sum_{k=1..n} (n+1)^(5*k)/k! * a(n-k) for n > 0 with a(0) = 1.
a(n) = A342202(5,n+1) = Sum_{r=1..(n+1)} (-1)^(r-1) * Sum_{s_1, ..., s_r} (1/(Product_{j=1..r} s_j!)) * Product_{j=1..r} (Sum_{i=1..j} s_i)^(5*s_j)), where the second sum is over lists (s_1, ..., s_r) of positive integers s_i such that Sum_{i=1..r} s_i = n+1. (Thus, the second sum is over all compositions of n+1. See Michel Marcus's PARI program in A342202.) - Petros Hadjicostas, Mar 10 2021

A342405 a(n) = (27^n - 3*9^n - 3*12^n)/6 + 6^n.

Original entry on oeis.org

0, 45, 2268, 76221, 2245320, 62858025, 1723364748, 46836754821, 1268169391440, 34282547074305, 926123262507828, 25011175461289821, 675371104361586360, 18235844869321055385, 492377645105637260508, 13294313813660319607221, 358947876218708733778080
Offset: 1

Views

Author

Petros Hadjicostas, Mar 10 2021

Keywords

Links

Crossrefs

Column 3 of A342202.

Formula

From Chai Wah Wu, Mar 11 2021: (Start)
a(n) = 54*a(n-1) - 963*a(n-2) + 6966*a(n-3) - 17496*a(n-4) for n > 4.
G.f.: 9*x^2*(-324*x^2 - 18*x + 5)/((6*x - 1)*(9*x - 1)*(12*x - 1)*(27*x - 1)). (End)
a(n) == 0 (mod 81) for n >= 3. - Hugo Pfoertner, Mar 11 2021
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