A300589
a(n) = A300617(n) / (n*(n+1)/2) for n>=1.
Original entry on oeis.org
1, 1, 5, 55, 1025, 28638, 1117831, 58157100, 3895841625, 327054041995, 33660663702514, 4170641243258042, 612634528823952155, 105303950053511041900, 20943400410601239618360, 4772694556432364600596272, 1235587041134996933696367753, 360653856192923791041427500825, 117894515649092645422159124253775, 42901062533218086978322192560871705
Offset: 1
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{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1) ); polcoeff( log(Ser(A)), n) / (n*(n+1)/2)}
for(n=1, 20, print1(a(n), ", "))
A300616
E.g.f. A(x) satisfies: [x^n] A(x)^n = n^2 * [x^(n-1)] A(x)^n for n>=1.
Original entry on oeis.org
1, 1, 7, 199, 14065, 1924201, 445859911, 161145717727, 85790577700129, 64427620614173425, 65943035132156264071, 89425725156530626400791, 156922032757769223085752337, 349233620942232034199096926489, 968890106809715834110637461124935, 3301188169350221687517822373590448111, 13634136452997022097853039839798901714241
Offset: 0
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 199*x^3/3! + 14065*x^4/4! + 1924201*x^5/5! + 445859911*x^6/6! + 161145717727*x^7/7! + 85790577700129*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^n begins:
n=1: [(1), (1), 7/2, 199/6, 14065/24, 1924201/120, 445859911/720, ...];
n=2: [1, (2), (8), 220/3, 3752/3, 502114/15, 57409744/45, ...];
n=3: [1, 3, (27/2), (243/2), 16035/8, 2098161/40, 157765131/80, ...];
n=4: [1, 4, 20, (536/3), (8576/3), 1096868/15, 121987336/45, ...];
n=5: [1, 5, 55/2, 1475/6, (91825/24), (2295625/24), 503279435/144, ...];
n=6: [1, 6, 36, 324, 4920, (601074/5), (21638664/5), 7491519768/35...];
n=7: [1, 7, 91/2, 2485/6, 147721/24, 17641687/120, (3752979139/720), (183895977811/720), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 8 = 2^2*2; 243/2 = 3^2*27/2; 8576/3 = 4^2*536/3; ...
illustrating that: [x^n] A(x)^n = n^2 * [x^(n-1)] A(x)^n.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 3*x^2 + 30*x^3 + 550*x^4 + 15375*x^5 + 601398*x^6 + 31299268*x^7 + 2093655600*x^8 + 175312873125*x^9 + 17987972309725*x^10 + ... + A300617(n)*x^n + ...
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{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1) ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))
A300619
O.g.f. A(x) satisfies: [x^n] exp( n * A(x) ) = n^3 * [x^(n-1)] exp( n * A(x) ) for n>=1.
Original entry on oeis.org
1, 7, 207, 14226, 1852800, 409408077, 142286748933, 73448832515952, 53835885818473473, 54041298732304775000, 72129250579997923194091, 124900802377559946754633602, 274851919918333747166200590840, 755158633069275870471471631726803, 2551279948230221759814139760682442500
Offset: 1
O.g.f.: A(x) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + 142286748933*x^7 + 73448832515952*x^8 + 53835885818473473*x^9 + ...
where
exp(A(x)) = 1 + x + 15*x^2/2! + 1285*x^3/3! + 347065*x^4/4! + 224232501*x^5/5! + 296201195791*x^6/6! + 719274160258585*x^7/7! + ... + A300618(n)*x^n/n! + ...
such that: [x^n] exp( n * A(x) ) = n^3 * [x^(n-1)] exp( n * A(x) ).
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{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1) ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))
A300620
Table of row functions R(n,x) that satisfy: [x^k] exp( k * R(n,x) ) = k^n * [x^(k-1)] exp( k * R(n,x) ) for k>=1, n>=1, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 3, 3, 1, 7, 30, 14, 1, 15, 207, 550, 85, 1, 31, 1290, 14226, 15375, 621, 1, 63, 7803, 340550, 1852800, 601398, 5236, 1, 127, 46830, 8086594, 215528250, 409408077, 31299268, 49680, 1, 255, 280647, 192663030, 25359510515, 280823532696, 142286748933, 2093655600, 521721, 1, 511, 1682130, 4605331346, 3013207159725, 197431364485587, 676005054191880, 73448832515952, 175312873125, 5994155
Offset: 1
This table of coefficients T(n,k) begins:
n=1: [1, 1, 3, 14, 85, 621, 5236, ...];
n=2: [1, 3, 30, 550, 15375, 601398, 31299268, ...];
n=3: [1, 7, 207, 14226, 1852800, 409408077, 142286748933, ...];
n=4: [1, 15, 1290, 340550, 215528250, 280823532696, 676005054191880, ...];
n=5: [1, 31, 7803, 8086594, 25359510515, 197431364485587, ...];
n=6: [1, 63, 46830, 192663030, 3013207159725, 140620832995924134, ...];
n=7: [1, 127, 280647, 4605331346, 359881205186350, 100749338488125315273, 82972785219971584775198767, ...]; ...
such that row functions R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfy:
[x^k] exp( k * R(n,x) ) = k^n * [x^(k-1)] exp( k * R(n,x) ) for k>=1.
Row functions R(n,x) begin:
R(1,x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 + ...
R(2,x) = x + 3*x^2 + 30*x^3 + 550*x^4 + 15375*x^5 + 601398*x^6 + 31299268*x^7 + ...
R(3,x) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + ...
R(4,x) = x + 15*x^2 + 1290*x^3 + 340550*x^4 + 215528250*x^5 + 280823532696*x^6 + ...
etc.
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{T(n,k) = my(A=[1]); for(i=1, k+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^n*V[#A-1] - V[#A])/(#A-1) ); polcoeff( log(Ser(A)), k)}
for(n=1, 8, for(k=1,8, print1(T(n,k), ", "));print(""))
A320414
O.g.f. A(x) satisfies: [x^n] exp(-n*A(x)) / (1 - n^2*x) = 0 for n >= 1.
Original entry on oeis.org
1, 5, 144, 10082, 1264535, 245875182, 68150171838, 25482852420656, 12358294214448753, 7544949396113515915, 5664150492647564303056, 5129806105907894893467492, 5516524251630079831171874313, 6950350571025359814277640201432, 10142321626803167978417939290871040, 16972169807104759800475307306764090752, 32287069027792648627362032680785777091413
Offset: 1
G.f.: A(x) = x + 5*x^2 + 144*x^3 + 10082*x^4 + 1264535*x^5 + 245875182*x^6 + 68150171838*x^7 + 25482852420656*x^8 + 12358294214448753*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) / (1 - n^2*x) begins:
n=1: [1, 0, -9, -862, -241719, -151666596, -176990046545, ...];
n=2: [1, 2, 0, -1616, -495232, -308287632, -357675223424, ...];
n=3: [1, 6, 87, 0, -693639, -475046118, -548265525633, ...];
n=4: [1, 12, 360, 14240, 0, -586861344, -749452608128, ...];
n=5: [1, 20, 975, 69430, 5820185, 0, -861802226825, ...];
n=6: [1, 30, 2136, 226368, 31268736, 4762503504, 0, ...];
n=7: [1, 42, 4095, 597044, 115492713, 27293766318, 6830535431671, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 11*x^2/2! + 895*x^3/3! + 245785*x^4/4! + 153050681*x^5/5! + 177988541251*x^6/6! + 344758772825671*x^7/7! + 1030280671456569905*x^8/8! + ...
exp(-A(x)) = 1 - x - 9*x^2/2! - 835*x^3/3! - 238271*x^4/4! - 150458001*x^5/5! - 176080046969*x^6/6! - 342201963425491*x^7/7! - 1024667102754203775*x^8/8! + ...
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{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m*x*Ser(A))/(1-m^2*x +x^2*O(x^m)))[m+1]/m ); A[n]}
for(n=1, 20, print1(a(n), ", "))
Showing 1-5 of 5 results.
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