A304321
Table of coefficients in row functions F'(n,x)/F(n,x) such that [x^k] exp( k^n * x ) / F(n,x) = 0 for k>=1 and n>=1.
Original entry on oeis.org
1, 1, 1, 1, 9, 1, 1, 49, 148, 1, 1, 225, 6877, 3493, 1, 1, 961, 229000, 1854545, 106431, 1, 1, 3969, 6737401, 612243125, 807478656, 3950832, 1, 1, 16129, 188580028, 172342090401, 3367384031526, 514798204147, 172325014, 1, 1, 65025, 5170118437, 45770504571813, 11657788116175751, 33056423981177346, 451182323794896, 8617033285, 1
Offset: 1
This table begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 9, 148, 3493, 106431, 3950832, 172325014, 8617033285, 485267003023, ...;
1, 49, 6877, 1854545, 807478656, 514798204147, 451182323794896, ...;
1, 225, 229000, 612243125, 3367384031526, 33056423981177346, ...;
1, 961, 6737401, 172342090401, 11657788116175751, 1722786509653595220757, ...;
1, 3969, 188580028, 45770504571813, 37854124915368647781, ...;
1, 16129, 5170118437, 11889402239702065, 120067639589726126102806, ...;
1, 65025, 140510362000, 3061712634885743125, 377436820462509018320487276, ...;
1, 261121, 3804508566001, 785701359968473902401, 1182303741240112494973150131501, ...; ...
Let F'(n,x)/F(n,x) denote the o.g.f. of row n of this table, then the coefficient of x^k in exp(k^n*x)/F(n,x) = 0 for k>=1 and n>=1.
-
m = 10(*rows*);
row[nn_] := Module[{F, s}, F = 1 + Sum[c[k] x^k, {k, m}]; s[n_] := Solve[ SeriesCoefficient[Exp[n^nn*x]/F, {x, 0, n}] == 0][[1]]; Do[F = F /. s[n], {n, m}]; CoefficientList[D[F, x]/F + O[x]^m, x]];
T = Array[row, m];
Table[T[[n-k+1, k]], {n, 1, m}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 27 2019 *)
-
{T(n,k) = my(A=[1],m); for(i=0, k, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^n +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[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),", "));)
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
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.
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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),", "))
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
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 + ...
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{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),", "))
A304314
Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^4 * x ) / F(x) = 0 for n>0.
Original entry on oeis.org
1, 225, 229000, 612243125, 3367384031526, 33056423981177346, 527146092112494861420, 12764850938355048224394925, 446065249480005516657138106375, 21615893741029073481369412949207860, 1406758471936562034421316174257309550136, 119755662436589797897149020637183902177930534
Offset: 0
O.g.f.: L(x) = 1 + 225*x + 229000*x^2 + 612243125*x^3 + 3367384031526*x^4 + 33056423981177346*x^5 + 527146092112494861420*x^6 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304324 :
F(x) = 1 + x + 113*x^2 + 76446*x^3 + 153143499*x^4 + 673638499100*x^5 + 5510097691767062*x^6 + 75312181798660695788*x^7 + ... + A304324(n)*x^n + ...
which satisfies [x^n] exp( n^4 * x ) / F(x) = 0 for n>0.
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m = 25;
F = 1 + Sum[c[k] x^k, {k, m}];
s[n_] := Solve[SeriesCoefficient[Exp[n^4*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 21 2018 *)
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{a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^4 +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}
for(n=0,25, print1( a(n),", "))
A304315
Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^5 * x ) / F(x) = 0 for n>0.
Original entry on oeis.org
1, 961, 6737401, 172342090401, 11657788116175751, 1722786509653595220757, 489506033977061086758261063, 243968979437942649897623460813009, 199025593654123221838381793032781035510, 251774439716905627952289102887999425054599511, 472942802381336010263584088374665504251010554412128, 1273071332950625956697135571575613091625334028239417955701
Offset: 0
O.g.f.: L(x) = 1 + 961*x + 6737401*x^2 + 172342090401*x^3 + 11657788116175751*x^4 + 1722786509653595220757*x^5 + 489506033977061086758261063*x^6 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304325 :
F(x) = 1 + x + 481*x^2 + 2246281*x^3 + 43087884081*x^4 + 2331601789103231*x^5 + 287133439746933073357*x^6 + 69929721774643572422651223*x^7 + ... + A304325(n)*x^n + ...
which satisfies [x^n] exp( n^5 * x ) / F(x) = 0 for n>0.
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m = 25;
F = 1 + Sum[c[k] x^k, {k, m}];
s[n_] := Solve[SeriesCoefficient[Exp[n^5*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 21 2018 *)
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{a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^4 +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}
for(n=0,25, print1( a(n),", "))
A006692
Number of connected n-state finite automata with 3 inputs.
Original entry on oeis.org
49, 6877, 1854545, 807478656, 514798204147, 451182323794896, 519961864703259753, 762210147961330421167, 1384945048774500147047194, 3055115321627096660341307614, 8043516699726480852467167758419, 24915939138210507189761922944830006, 89709850983809128394441772076036629240
Offset: 1
- Robert W. Robinson, Counting strongly connected finite automata, pages 671-685 in "Graph theory with applications to algorithms and computer science." Proceedings of the fifth international conference held at Western Michigan University, Kalamazoo, Mich., June 4-8, 1984. Edited by Y. Alavi, G. Chartrand, L. Lesniak [L. M. Lesniak-Foster], D. R. Lick and C. E. Wall. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. xv+810 pp. ISBN: 0-471-81635-3; Math Review MR0812651 (86g:05026).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Hugo Pfoertner, Table of n, a(n) for n = 1..200
- R. W. Robinson, Counting strongly connected finite automata, pages 671-685 in "Graph theory with applications to algorithms and computer science." Proceedings of the fifth international conference held at Western Michigan University, Kalamazoo, Mich., June 4-8, 1984. Edited by Y. Alavi, G. Chartrand, L. Lesniak [L. M. Lesniak-Foster], D. R. Lick and C. E. Wall. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. xv+810 pp. ISBN: 0-471-81635-3; Math Review MR0812651. (86g:05026). [Annotated scanned copy, with permission of the author.]
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