A304322 -id:A304322 - cp's OEIS frontend

A107668 Column 0 of triangle A107667.

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

Offset: 0

Paul D. Hanna, Jun 07 2005

nonn

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

			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)

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A107667, A107669, A107670, A052200.

Cf. A304322, A107675, A304394, A304395, A342202.

-- 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

-- 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

{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),", "))

/* 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

/* 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

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

A304320 Table of coefficients in row functions R(n,x) such that [x^k] exp( k^n * x ) / R(n,x) = 0 for k>=1 and n>=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 25, 54, 1, 1, 1, 113, 2317, 935, 1, 1, 1, 481, 76446, 466241, 22417, 1, 1, 1, 1985, 2246281, 153143499, 162016980, 685592, 1, 1, 1, 8065, 62861994, 43087884081, 673638499100, 85975473871, 25431764, 1, 1, 1, 32513, 1723380877, 11442690973075, 2331601789103231, 5510097691767062, 64545532370208, 1106630687, 1, 1, 1, 130561, 46836819846, 2972352315820441, 7570836550478960487, 287133439746933073357, 75312181798660695788, 65062315637060121, 55174867339, 1

Offset: 1

Paul D. Hanna, May 11 2018

It is striking that the coefficients in this table consist entirely of integers.

			This table begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 1, 5, 54, 935, 22417, 685592, 25431764, 1106630687, 55174867339, ...;
1, 1, 25, 2317, 466241, 162016980, 85975473871, 64545532370208, ...;
1, 1, 113, 76446, 153143499, 673638499100, 5510097691767062, ...;
1, 1, 481, 2246281, 43087884081, 2331601789103231, 287133439746933073357, ...;
1, 1, 1985, 62861994, 11442690973075, 7570836550478960487, ...;
1, 1, 8065, 1723380877, 2972352315820441, 24013530904194819396970, ...;
1, 1, 32513, 46836819846, 765428206086770699, 75487364859452767380638650, ...;
1, 1, 130561, 1268169652561, 196425341268811084961, 236460748444613412476233431261, ...; ...
Let R(n,x) denote the o.g.f. of row n of this table, then the coefficient of x^k in exp(k^n*x)/R(n,x) = 0 for k>=1 and n>=1.

Paul D. Hanna, Table of n, a(n) for n = 1..1326 as a flattened table read by antidiagonals 1..51.

Cf. A304321, A304322 (row 2), A304323 (row 3), A304324 (row 4), A304325 (row 5), A337551 (diagonal).

{T(n,k) = my(A=[1],m); for(i=1, k, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^n +x*O(x^m)) / Ser(A) )[m] ); A[k+1]}
/* Print table: */
for(n=1,8, for(k=0,8, print1( T(n,k),", "));print(""))
/* Print as a flattened table: */
for(n=0,10, for(k=0,n, print1( T(n-k+1,k),", "));)

For fixed row r > 1 is a(n) ~ sqrt(1-c) * r^(r*n) * n^((r-1)*n - 1/2) / (sqrt(2*Pi) * c^n * (r-c)^((r-1)*n) * exp((r-1)*n)), where c = -LambertW(-r*exp(-r)). - Vaclav Kotesovec, Aug 31 2020

A304312 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^2 * x ) / F(x) = 0 for n>0.

Original entry on oeis.org

1, 9, 148, 3493, 106431, 3950832, 172325014, 8617033285, 485267003023, 30363691715629, 2088698040637242, 156612539215405732, 12709745319947141220, 1109746209390479579732, 103724343230007402591558, 10332348604630683943445797, 1092720669631704348689818959, 122274820828415241343176467043, 14433472319311799728710020346232

Offset: 0

Paul D. Hanna, May 11 2018

nonn

Is this sequence essentially the same as A006691?
Conjecture: a(n) is the number of connected n-state finite automata with 2 inputs (A006691). [I believe the name of A006691 should be changed to read "(n+1)-state". See my comments in A006691. - Petros Hadjicostas, Feb 26 2021]
Equals row 2 of table A304321.

			O.g.f.: L(x) = 1 + 9*x + 148*x^2 + 3493*x^3 + 106431*x^4 + 3950832*x^5 + 172325014*x^6 + 8617033285*x^7 + 485267003023*x^8 + 30363691715629*x^9 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304322 :
F(x) = 1 + x + 5*x^2 + 54*x^3 + 935*x^4 + 22417*x^5 + 685592*x^6 + 25431764*x^7 + 1106630687*x^8 + 55174867339*x^9 + 3097872254493*x^10 + ... + A304322(n)*x^n + ...
which satisfies [x^n] exp( n^2 * x ) / F(x) = 0 for n>0.

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A304322, A006691, A304321, A304313, A304314, A304315.

m = 25;
F = 1 + Sum[c[k] x^k, {k, m}];
s[n_] := Solve[SeriesCoefficient[Exp[n^2 * x]/F, {x, 0, n}] == 0][[1]];
Do[F = F /. s[n], {n, m}];
CoefficientList[D[F, x]/F + O[x]^m, x] (* Jean-François Alcover, May 20 2018 *)

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

Logarithmic derivative of the o.g.f. of A304322.
For n>=1, a(n) = B_{n+1}((n+1)^2-0!*a(0),-1!*a(1),...,-(n-1)!*a(n-1),0) / n!, where B_{n+1}(...) is the (n+1)-st complete exponential Bell polynomial. - Max Alekseyev, Jun 18 2018
a(n) ~ sqrt(1-c) * 2^(2*n + 3/2) * n^(n + 3/2) / (sqrt(Pi) * c^(n+1) * (2-c)^(n+1) * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

A304324 O.g.f. A(x) satisfies: [x^n] exp( n^4 * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 1, 113, 76446, 153143499, 673638499100, 5510097691767062, 75312181798660695788, 1595682359653020033714019, 49564410138113345565513815041, 2161639124039437373346491749452440, 127889301139607880711208251726358504898, 9979766671875039854419652569806336108694074

Offset: 0

Paul D. Hanna, May 11 2018

nonn

It is conjectured that the coefficients of o.g.f. A(x) consist entirely of integers.
Equals row 4 of table A304320.
O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A304394.
Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304314.
Conjecture: given o.g.f. A(x), the coefficient of x^n in A'(x)/A(x) is the number of connected n-state finite automata with 4 inputs.

			O.g.f.: A(x) = 1 + x + 113*x^2 + 76446*x^3 + 153143499*x^4 + 673638499100*x^5 + 5510097691767062*x^6 + 75312181798660695788*x^7 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^4*x) / A(x) begins:
n=0: [1, -1, -224, -457326, -3671476224, -80797824300000, ...];
n=1: [1, 0, -225, -458000, -3673306875, -80816186256624, ...];
n=2: [1, 15, 0, -464750, -3701040000, -81092721606624, ...];
n=3: [1, 80, 6175, 0, -3787546875, -82312696206624, ...];
n=4: [1, 255, 64800, 15951250, 0, -84756571206624, ...];
n=5: [1, 624, 389151, 242091424, 146271536901, 0, ...];
n=6: [1, 1295, 1676800, 2170415250, 2804103120000, 3524906587193376, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^4*x ) / A(x) = 0 for n>=0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304314:
A'(x)/A(x) = 1 + 225*x + 229000*x^2 + 612243125*x^3 + 3367384031526*x^4 + 33056423981177346*x^5 + 527146092112494861420*x^6 + ... + A304314(n)*x^n + ...
INVERT TRANSFORM.
1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A304394:
B(x) = 1 + 112*x + 76221*x^2 + 152978176*x^3 + 673315202500*x^4 + 5508710472669120*x^5 + 75300988091046198131*x^6 + ... + A304394(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A304320, A304314, A304321, A304322, A304323, A304325.

Cf. A304394.

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

a(n) ~ sqrt(1-c) * 4^(4*n) * n^(3*n - 1/2) / (sqrt(2*Pi) * c^n * (4-c)^(3*n) * exp(3*n)), where c = -LambertW(-4*exp(-4)). - Vaclav Kotesovec, Aug 31 2020

A304323 O.g.f. A(x) satisfies: [x^n] exp( n^3 * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 1, 25, 2317, 466241, 162016980, 85975473871, 64545532370208, 65062315637060121, 84756897268784533255, 138581022247955235150982, 277878562828788369685779910, 670574499099019193091230751539, 1917288315895234006935990419270242, 6409780596355519454337664637246378856, 24774712941456386970945752104780461007848, 109632095120643795798521114315908854415860345

Offset: 0

Paul D. Hanna, May 11 2018

nonn

It is conjectured that the coefficients of o.g.f. A(x) consist entirely of integers.
Equals row 3 of table A304320.
O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A107675.
Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304312.
Conjecture: given o.g.f. A(x), the coefficient of x^n in A'(x)/A(x) is the number of connected n-state finite automata with 3 inputs (A006692).

			O.g.f.: A(x) = 1 + x + 25*x^2 + 2317*x^3 + 466241*x^4 + 162016980*x^5 + 85975473871*x^6 + 64545532370208*x^7 + 65062315637060121*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^3*x) / A(x) begins:
n=0: [1, -1, -48, -13608, -11065344, -19317285000, -61649646030720, ...];
n=1: [1, 0, -49, -13754, -11120067, -19372748284, -61765715993765, ...];
n=2: [1, 7, 0, -14440, -11517184, -19768841352, -62587640670464, ...];
n=3: [1, 26, 627, 0, -12292251, -20908064898, -64905483973113, ...];
n=4: [1, 63, 3920, 227032, 0, -22551552136, -69768485886848, ...];
n=5: [1, 124, 15327, 1874642, 213958781, 0, -75806801733845, ...];
n=6: [1, 215, 46176, 9893016, 2100211968, 416846973816, 0, ...];
n=7: [1, 342, 116915, 39937660, 13616254341, 4604681316698, 1458047845980391, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^3*x ) / A(x) = 0 for n>=0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304313:
A'(x)/A(x) = 1 + 49*x + 6877*x^2 + 1854545*x^3 + 807478656*x^4 + 514798204147*x^5 + 451182323794896*x^6 + 519961864703259753*x^7 + ... + A304313(n)*x^n +...
INVERT TRANSFORM.
1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A107675:
B(x) = 1 + 24*x + 2268*x^2 + 461056*x^3 + 160977375*x^4 + 85624508376*x^5 + 64363893844726*x^6 + ... + A107675(n)*x^n + ...

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

Cf. A304320, A304313, A304321, A304322, A304324, A304325.

Cf. A006691, A107675.

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^3 +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0,25, print1( a(n),", "))

a(n) ~ sqrt(1-c) * 3^(3*n) * n^(2*n - 1/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n) * exp(2*n)), where c = -A226750 = -LambertW(-3*exp(-3)). - Vaclav Kotesovec, Aug 31 2020

A304325 O.g.f. A(x) satisfies: [x^n] exp( n^5 * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 1, 481, 2246281, 43087884081, 2331601789103231, 287133439746933073357, 69929721774643572422651223, 30496192503451926066104677123329, 22113985380962062942048847693898939310, 25177466100486219354624677349405490885006591, 42994825404638061265611776726882581676486680632128

Offset: 0

Paul D. Hanna, May 11 2018

nonn

It is conjectured that the coefficients of o.g.f. A(x) consist entirely of integers.
Equals row 5 of table A304320.
O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A304395.
Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304315.
Conjecture: given o.g.f. A(x), the coefficient of x^n in A'(x)/A(x) enumerates the connected n-state finite automata with 5 inputs.

			O.g.f.: A(x) = 1 + x + 481*x^2 + 2246281*x^3 + 43087884081*x^4 + 2331601789103231*x^5 + 287133439746933073357*x^6 + 69929721774643572422651223*x^7 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^5*x) / A(x) begins:
n=0: [1, -1, -960, -13471920, -1033995878400, -279781615181250000, ...];
n=1: [1, 0, -961, -13474802, -1034049771843, -279786785295370804, ...];
n=2: [1, 31, 0, -13534384, -1035725264896, -279947192760516048, ...];
n=3: [1, 242, 57603, 0, -1044001318107, -281045183102366562, ...];
n=4: [1, 1023, 1045568, 1054175056, 0, -284106842971323856, ...];
n=5: [1, 3124, 9758415, 30465809330, 93986716449725, 0, ...];
n=6: [1, 7775, 60449664, 469967719248, 3652476388472832, 28079364132086235696, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^5*x ) / A(x) = 0 for n >= 0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304315:
A'(x)/A(x) = 1 + 961*x + 6737401*x^2 + 172342090401*x^3 + 11657788116175751*x^4 + 1722786509653595220757*x^5 + 489506033977061086758261063*x^6 + ... + A304315(n)*x^n +...
INVERT TRANSFORM.
1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A304395:
B(x) = 1 + 480*x + 2245320*x^2 + 43083161600*x^3 + 2331513459843750*x^4 + 287128730182879382976*x^5 + ... + A304395(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A304320, A304315, A304321, A304322, A304323, A304324.

Cf. A304395.

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^5 +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0,25, print1( a(n),", "))

a(n) ~ sqrt(1-c) * 5^(5*n) * n^(4*n - 1/2) / (sqrt(2*Pi) * c^n * (5-c)^(4*n) * exp(4*n)), where c = -LambertW(-5*exp(-5)). - Vaclav Kotesovec, Aug 31 2020

A304863 O.g.f. A(x) satisfies: [x^n] exp( n(n+2) x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 3, 17, 180, 3079, 74271, 2308940, 87438684, 3888682559, 198073751505, 11348409001233, 721483807171188, 50361931297722244, 3827114191186713588, 314413091556481490640, 27761835704580647457012, 2621495363274661266785679, 263593068966612639018287637, 28117066903131481643928647363, 3171150259810035292799245555884, 377044852592342586608552585592079

Offset: 0

Paul D. Hanna, May 19 2018

nonn

			O.g.f.: A(x) = 1 + 3*x + 17*x^2 + 180*x^3 + 3079*x^4 + 74271*x^5 + 2308940*x^6 + 87438684*x^7 + 3888682559*x^8 + 198073751505*x^9 + 11348409001233*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+2)*x) / A(x) begins:
n=0: [1, -3, -16, -630, -50112, -6665400, -1325134080, -366674636160, ...];
n=1: [1, 0, -25, -828, -58779, -7479072, -1452239685, -395811449100, ...];
n=2: [1, 5, 0, -1078, -78464, -9183672, -1700942720, -450843184000, ...];
n=3: [1, 12, 119, 0, -99387, -12381300, -2151101205, -544666984560, ...];
n=4: [1, 21, 416, 6858, 0, -15533496, -2923952256, -703585823616, ...];
n=5: [1, 32, 999, 29540, 730213, 0, -3653179205, -962999633260, ...];
n=6: [1, 45, 2000, 86922, 3589056, 124275528, 0, -1200826684800, ...];
n=7: [1, 60, 3575, 210672, 12162501, 668679228, 30900268395, 0, ...];
n=8: [1, 77, 5904, 449930, 33949888, 2513449800, 177544721920, 10559736679040, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+2)*x ) / A(x) = 0 for n>=0.
RELATED SERIES.
The logarithmic derivative of A(x) yields:
A'(x)/A(x) = 3 + 25*x + 414*x^2 + 10109*x^3 + 320253*x^4 + 12346720*x^5 + 557708406*x^6 + 28786882117*x^7 + 1668054884229*x^8 + 107077380781005*x^9 + ...
1 - 1/A(x) = 3*x + 8*x^2 + 105*x^3 + 2088*x^4 + 55545*x^5 + 1840464*x^6 + 72752904*x^7 + 3334122880*x^8 + 173569203225*x^9 + 10108800765000*x^10 + ...

Cf. A304322, A304318, A304319, A304864, A304865.

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

A304864 O.g.f. A(x) satisfies: [x^n] exp( n(n+3) x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 4, 26, 288, 5012, 122608, 3869456, 148838816, 6721823600, 347434618432, 20180665251360, 1299399587904384, 91769604540962816, 7049102617933604352, 584848346900868001792, 52109481481410100183552, 4961586770799906448318208, 502707358017324652042259456, 54000226687663791374322245120, 6129804668943947684749062516736, 733179029209444818691965317379072

Offset: 0

Paul D. Hanna, May 19 2018

nonn

			O.g.f.: A(x) = 1 + 4*x + 26*x^2 + 288*x^3 + 5012*x^4 + 122608*x^5 + 3869456*x^6 + 148838816*x^7 + 6721823600*x^8 + 347434618432*x^9 + 20180665251360*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+3)*x) / A(x) begins:
n=0: [1, -4, -20, -864, -72576, -10060800, -2068882560, -589017945600, ...];
n=1: [1, 0, -36, -1232, -89088, -11667456, -2328963200, -650497926144, ...];
n=2: [1, 6, 0, -1664, -125136, -14853600, -2803074560, -757869964800, ...];
n=3: [1, 14, 160, 0, -162000, -20768352, -3651775488, -937273259520, ...];
n=4: [1, 24, 540, 10000, 0, -26468352, -5107476608, -1241737082880, ...];
n=5: [1, 36, 1260, 41536, 1133184, 0, -6460818560, -1740188582400, ...];
n=6: [1, 50, 2464, 118368, 5374512, 202760544, 0, -2192436486144, ...];
n=7: [1, 66, 4320, 279136, 17619504, 1054101600, 52553405440, 0, ...];
n=8: [1, 84, 7020, 582400, 47760000, 3832731648, 292170316672, 18603667330560, 0, ...]; ...
RELATED SERIES.
The logarithmic derivative of A(x) yields:
A'(x)/A(x) = 4 + 36*x + 616*x^2 + 15496*x^3 + 504624*x^4 + 19947072*x^5 + 921521248*x^6 + 48536700064*x^7 + 2864002270720*x^8 + 186878075521216*x^9 + ...
1 - 1/A(x) = 4*x + 10*x^2 + 144*x^3 + 3024*x^4 + 83840*x^5 + 2873448*x^6 + 116868640*x^7 + 5488631808*x^8 + 291890096640*x^9 + 17321970359200*x^10 + ...

Cf. A304322, A304318, A304319, A304863, A304865.

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

A304865 O.g.f. A(x) satisfies: [x^n] exp( n(n+4) x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 5, 37, 434, 7751, 193101, 6200872, 242667316, 11144759839, 585318934391, 34511211188637, 2253285826341378, 161201686356627524, 12530568505972885004, 1051099249634285619168, 94603882448795669308980, 9092091650779263675187695, 929177036869575506758681035, 100608724821944458615599713935, 11504982932704269804549116593702, 1385525417578463389730054278506959

Offset: 0

Paul D. Hanna, May 19 2018

nonn

			O.g.f.: A(x) = 1 + 5*x + 37*x^2 + 434*x^3 + 7751*x^4 + 193101*x^5 + 6200872*x^6 + 242667316*x^7 + 11144759839*x^8 + 585318934391*x^9 + 34511211188637*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+4)*x) / A(x) begins:
n=0: [1, -5, -24, -1134, -100608, -14542200, -3095496000, -907608905280, ...];
n=1: [1, 0, -49, -1744, -128763, -17383400, -3572628125, -1024052930280, ...];
n=2: [1, 7, 0, -2430, -189600, -22895928, -4410982656, -1218708054720, ...];
n=3: [1, 16, 207, 0, -250107, -33107544, -5910144669, -1540910769048, ...];
n=4: [1, 27, 680, 13970, 0, -42775928, -8486494016, -2090851421760, ...];
n=5: [1, 40, 1551, 56376, 1681797, 0, -10852876125, -2994692165280, ...];
n=6: [1, 55, 2976, 156546, 7748832, 316211400, 0, -3807596825280, ...];
n=7: [1, 72, 5135, 360920, 24718725, 1597879072, 85448027299, 0, ...];
n=8: [1, 91, 8232, 738450, 65376768, 5650680456, 462123838656, 31350065660352, 0, ...]; ...
RELATED SERIES.
The logarithmic derivative of A(x) yields:
A'(x)/A(x) = 5 + 49*x + 872*x^2 + 22661*x^3 + 759915*x^4 + 30843448*x^5 + 1459277062*x^6 + 78529473925*x^7 + 4724556111179*x^8 + 313739794874469*x^9 + ...
1 - 1/A(x) = 5*x + 12*x^2 + 189*x^3 + 4192*x^4 + 121185*x^5 + 4299300*x^6 + 180081132*x^7 + 8675950464*x^8 + 471853727865*x^9 + 28563862383700*x^10 + ...

Cf. A304322, A304318, A304319, A304863, A304864.

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m+3) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

cp's OEIS Frontend

A107668 Column 0 of triangle A107667.

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A304320 Table of coefficients in row functions R(n,x) such that [x^k] exp( k^n * x ) / R(n,x) = 0 for k>=1 and n>=1.

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A304312 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^2 * x ) / F(x) = 0 for n>0.

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A304324 O.g.f. A(x) satisfies: [x^n] exp( n^4 * x ) / A(x) = 0 for n>0.

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A304323 O.g.f. A(x) satisfies: [x^n] exp( n^3 * x ) / A(x) = 0 for n>0.

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A304325 O.g.f. A(x) satisfies: [x^n] exp( n^5 * x ) / A(x) = 0 for n>0.

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A304863 O.g.f. A(x) satisfies: [x^n] exp( n*(n+2) * x ) / A(x) = 0 for n>0.

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A304863 O.g.f. A(x) satisfies: [x^n] exp( n(n+2) x ) / A(x) = 0 for n>0.

A304864 O.g.f. A(x) satisfies: [x^n] exp( n(n+3) x ) / A(x) = 0 for n>0.

A304865 O.g.f. A(x) satisfies: [x^n] exp( n(n+4) x ) / A(x) = 0 for n>0.