A337458 -id:A337458 - cp's OEIS frontend

A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

4, 0, 6, 3, 7, 5, 7, 3, 9, 9, 5, 9, 9, 5, 9, 9, 0, 7, 6, 7, 6, 9, 5, 8, 1, 2, 4, 1, 2, 4, 8, 3, 9, 7, 5, 8, 2, 1, 0, 9, 9, 7, 5, 7, 5, 1, 8, 1, 1, 4, 0, 6, 3, 5, 0, 0, 0, 4, 9, 5, 4, 8, 8, 3, 0, 3, 9, 1, 5, 0, 1, 5, 1, 8, 3, 8, 1, 2, 0, 4, 9, 7, 6, 7, 2, 5, 0, 0, 7, 2, 3, 3, 8, 1, 5, 5, 9, 2, 8, 5, 8, 2, 9, 3, 8

Offset: 0

José María Grau Ribas, Jun 17 2013

			-0.4063757399599599076769581241248397582109975751811406350004954883....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A106533, A187655, A187657, A217899, A217900, A226750, A243227, A245109, A247238, A258467, A337458.

RealDigits[N[ProductLog[-2/E^2], 105]][[1]] (* corrected by Vaclav Kotesovec, Feb 21 2014 *)

solve(x=-1, x=0, x*exp(x) + 2*exp(-2)) \\ G. C. Greubel, Nov 15 2017

Equals -2*A106533.

Equals LambertW(-2*exp(-2)).

A337457 O.g.f. A(x) satisfies: [x^n] exp( n(n-1) x/A(x) ) = 0 for n > 0.

Original entry on oeis.org

1, 1, 1, 7, 93, 1859, 49357, 1629227, 64149805, 2929386667, 152027131261, 8830653890299, 567303319553421, 39924294419453931, 3053895154472856285, 252244319795920299419, 22373037117819632459821, 2120745476831765696381387, 213946972632171665440620925, 22887117259538879173402222075

Offset: 0

Paul D. Hanna, Aug 28 2020

nonn

It is remarkable that this sequence consists entirely of integers.

			O.g.f.: A(x) = 1 + x + x^2 + 7*x^3 + 93*x^4 + 1859*x^5 + 49357*x^6 + 1629227*x^7 + 64149805*x^8 + 2929386667*x^9 + 152027131261*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n*(n-1)*x/A(x) ) begins:
n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
n=2: [1, 2, 0, -16, -320, -21888, -2648576, -494325760, ...];
n=3: [1, 6, 24, 0, -1728, -88704, -9621504, -1715198976, ...];
n=4: [1, 12, 120, 864, 0, -281088, -26873856, -4328017920, ...];
n=5: [1, 20, 360, 5600, 65920, 0, -66944000, -10207436800, ...];
n=6: [1, 30, 840, 21600, 492480, 8784000, 0, -22098355200, ...];
n=7: [1, 42, 1680, 63504, 2237760, 71229312, 1814690304, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n-1)*x/A(x) ) = 0 for n>0.
RELATED SERIES.
Define B(x) = A(x*B(x)), which begins
B(x) = 1 + x + 2*x^2 + 11*x^3 + 130*x^4 + 2450*x^5 + 63012*x^6 + 2040779*x^7 + 79377914*x^8 + 3594766694*x^9 + ... + A337458(n)*x^n + ...
then the table of coefficients of x^k/k! in exp(n*(n+1)*x) / B(x)^(n+1) begins:
n=0: [1, -1, -2, -48, -2616, -262080, -41718240, -9630270720, ...];
n=1: [1, 0, -6, -112, -5592, -547968, -86345120, -19809990912, ...];
n=2: [1, 3, 0, -222, -10728, -958824, -144971712, -32519314080, ...];
n=3: [1, 8, 52, 0, -18648, -1693248, -236690784, -50727983616, ...];
n=4: [1, 15, 210, 2420, 0, -2739720, -399251600, -80125144800, ...];
n=5: [1, 24, 558, 12192, 221184, 0, -616918320, -131299591680, ...];
n=6: [1, 35, 1204, 40278, 1272768, 33597312, 0, -196436730672, ...];
n=7: [1, 48, 2280, 106688, 4869552, 210771456, 7654459648, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+1)*x ) / B(x)^(n+1) = 0 for n>0.
Also note that B(x) = (1/x)*Series_Reversion( x/A(x) ) and A(x) = B(x/A(x)).

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

Cf. A337458.

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

Given o.g.f. A(x), define B(x) = A(x*B(x)), then B(x) is the o.g.f. of A337458 and satisfies [x^n] exp( n*(n+1)*x ) / B(x)^(n+1) = 0 for n>0.

a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.083103344220475784... - Vaclav Kotesovec, Aug 31 2020

A337576 O.g.f. A(x) satisfies: [x^n] exp( n^2(n+1) x ) / A(x)^(n+1) = 0 for n > 0.

Original entry on oeis.org

1, 1, 14, 947, 157190, 47437866, 22437363324, 15246207565643, 14053536511674526, 16868801353366004990, 25566893078760354005252, 47761059837097334197007118, 107843046053558916525978556156, 289613430019820775682179202404084

Offset: 0

Paul D. Hanna, Sep 02 2020

nonn

It is remarkable that this sequence consists entirely of integers.

			O.g.f.: A(x) = 1 + x + 14*x^2 + 947*x^3 + 157190*x^4 + 47437866*x^5 + 22437363324*x^6 + 15246207565643*x^7 + 14053536511674526*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2*(n+1)*x ) / A(x)^(n+1) begins:
n=0: [1, -1, -26, -5520, -3723384, -5652041280, -16083171669600, ...];
n=1: [1, 0, -54, -11200, -7486872, -11338403328, -32230618603040, ...];
n=2: [1, 9, 0, -18258, -11861352, -17522277048, -49272492906432, ...];
n=3: [1, 32, 916, 0, -17438424, -25288921344, -69043257103968, ...];
n=4: [1, 75, 5490, 363500, 0, -35101453320, -94993441197200, ...];
n=5: [1, 144, 20574, 2882400, 368064576, 0, -127110906431280, ...];
n=6: [1, 245, 59836, 14528010, 3470388768, 759773089152, 0, ...];
n=7: [1, 384, 147240, 56329472, 21453513648, 8058471570432, 2785824326725888, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^2*(n+1)*x ) / A(x)^(n+1) = 0 for n>0.
RELATED SERIES.
Define B(x) = A(x/B(x)), which begins
B(x) = 1 + x + 13*x^2 + 907*x^3 + 153145*x^4 + 46602295*x^5 + 22140651001*x^6 + 15084920403375*x^7 + ... + A337575(n)*x^n + ...
then the table of coefficients of x^k/k! in exp( n*(n-1)^2 * x/B(x) ) begins:
n=0: [1, 0, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 0, 0, 0, 0, 0, 0, 0, ...];
n=2: [1, 2, 0, -160, -43520, -36711168, -67072065536, ...];
n=3: [1, 12, 120, 0, -293760, -234067968, -415963247616, ...];
n=4: [1, 36, 1224, 36288, 0, -792405504, -1355831322624, ...];
n=5: [1, 80, 6240, 467840, 31356160, 0, -3403785728000, ...];
n=6: [1, 150, 22200, 3229200, 456364800, 58514400000, 0, ...];
n=7: [1, 252, 63000, 15603840, 3817860480, 913835768832, 200316485182464, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n-1)^2 * x/B(x) ) = 0 for n>0.
Also note that B(x) = x/Series_Reversion( x*A(x) ) and A(x) = B(x*A(x)).

Cf. A337575, A337458, A337578.

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

Given o.g.f. A(x), define B(x) = A(x/B(x)), then B(x) is the o.g.f. of A337575 and satisfies [x^n] exp( n*(n-1)^2 * x/B(x) ) = 0 for n>0.

A337578 O.g.f. A(x) satisfies: [x^n] exp( n(n+1)^2 x ) / A(x)^(n+1) = 0 for n > 0.

Original entry on oeis.org

1, 2, 26, 1756, 301140, 94035272, 45829458720, 31938032357440, 30067763435823664, 36733573872360057568, 56505395303074211175584, 106885908721218962904619840, 243929162811364027276490748288, 661085501644539366377254535077376

Offset: 0

Paul D. Hanna, Sep 02 2020

nonn

It is remarkable that this sequence consists entirely of integers.

			O.g.f.: A(x) = 1 + 2*x + 26*x^2 + 1756*x^3 + 301140*x^4 + 94035272*x^5 + 45829458720*x^6 + 31938032357440*x^7 + 30067763435823664*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n*(n+1)^2*x ) / A(x)^(n+1) begins:
n=0: [1, -2, -44, -9960, -7049664, -11131647360, -32715852151680, ...];
n=1: [1, 0, -96, -20480, -14247072, -22395261696, -65687348011520, ...];
n=2: [1, 12, 0, -34176, -22928112, -34905615552, -100977330265344, ...];
n=3: [1, 40, 1408, 0, -34275648, -51114811392, -142803802229760, ...];
n=4: [1, 90, 7860, 613000, 0, -71887626240, -199085724252800, ...];
n=5: [1, 168, 27936, 4535040, 663960096, 0, -269327647065600, ...];
n=6: [1, 280, 78064, 21598080, 5858601168, 1443397611264, 0, ...];
n=7: [1, 432, 186240, 80041984, 34200321408, 14371727121408, 5514496883009536, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+1)^2*x ) / A(x)^(n+1) = 0 for n>0.
RELATED SERIES.
Define B(x) = A(x/B(x)), which begins
B(x) = 1 + 2*x + 22*x^2 + 1616*x^3 + 286700*x^4 + 90914400*x^5 + 44673096808*x^6 + 31286975152640*x^7 + ... + A337577(n)*x^n + ...
then the table of coefficients of x^k/k! in exp( n*(n-1)^2 * x/B(x) ) begins:
n=0: [1, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 0, 0, 0, 0, 0, 0, ...];
n=2: [1, 4, 0, -560, -154880, -137342976, -261610747904, ...];
n=3: [1, 18, 252, 0, -822960, -670328352, -1230620630976, ...];
n=4: [1, 48, 2112, 77760, 0, -2077949952, -3628874151936, ...];
n=5: [1, 100, 9600, 869200, 68473600, 0, -8724419840000, ...];
n=6: [1, 180, 31680, 5423760, 890714880, 130187520000, 0, ...];
n=7: [1, 294, 85260, 24343200, 6817260240, 1850897137824, 453595543361856, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n-1)^2 * x/B(x) ) = 0 for n>0.
Also note that B(x) = x/Series_Reversion( x*A(x) ) and A(x) = B(x*A(x)).

Cf. A337577, A337576, A337458, A337579.

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

Given o.g.f. A(x), define B(x) = A(x/B(x)), then B(x) is the o.g.f. of A337577 and satisfies [x^n] exp( n^2*(n-1) * x/B(x) ) = 0 for n>0.

A337579 O.g.f. A(x) satisfies: [x^n] exp( n(n+1)^2 x ) / A(x)^((n+1)^2) = 0 for n>0.

Original entry on oeis.org

1, 1, 5, 202, 25741, 6481768, 2661785172, 1606979708104, 1336018641201031, 1461946920710738032, 2036450966030220362632, 3519187269661662800713808, 7390652429852470066011519746, 18545709306030399397877283499248, 54823008100459892066683079355901888, 188621839026471088419358039473633535392

Offset: 0

Paul D. Hanna, Sep 01 2020

nonn

It is remarkable that this sequence consists entirely of integers.

			O.g.f.: A(x) = 1 + x + 5*x^2 + 202*x^3 + 25741*x^4 + 6481768*x^5 + 2661785172*x^6 + 1606979708104*x^7 + 1336018641201031*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+1)^2*x) / A(x)^((n+1)^2) begins:
n=0: [1, -1, -8, -1158, -607824, -771471360, -1906996245120, ...];
n=1: [1, 0, -36, -4736, -2447112, -3096809856, -7645376634080, ...];
n=2: [1, 9, 0, -12114, -5911488, -7219467792, -17580593299968, ...];
n=3: [1, 32, 880, 0, -12002784, -14133084672, -33100636472064, ...];
n=4: [1, 75, 5400, 341650, 0, -25227867600, -57875848640000, ...];
n=5: [1, 144, 20412, 2803392, 343375416, 0, -95154559008480, ...];
n=6: [1, 245, 59584, 14323974, 3357877488, 709290480864, 0, ...];
n=7: [1, 384, 146880, 55883776, 21079051392, 7789007628288, 2612787154865152, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp(n*(n+1)^2*x) / A(x)^((n+1)^2) = 0 for n>0.
RELATED SERIES.
log(A(x)) = x + 9*x^2/2 + 592*x^3/3 + 102125*x^4/4 + 32276196*x^5/5 + 15931091190*x^6/6 + 11230009495552*x^7/7 + 10675195543084221*x^8/8 + ...
where [x^n] exp( (n+1)^2 * (n*x - log(A(x))) ) = 0 for n>0.

Cf. A337458.

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

A383559 O.g.f. A(x) satisfies: [x^n] exp( n(2n+1)*x ) / A(x) = 0 for n > 0.

Original entry on oeis.org

1, 3, 29, 609, 20857, 997671, 61114409, 4548317073, 397323349505, 39774233809179, 4483232458612245, 561425116837715457, 77289022946177141161, 11597365849594347661839, 1883429636306366952452433, 329083700898584984268782241, 61549497773760817234065857793, 12268604214374346472111552473267

Offset: 0

Paul D. Hanna, May 17 2025

nonn

			O.g.f.: A(x) = 1 + 3*x + 29*x^2 + 609*x^3 + 20857*x^4 + 997671*x^5 + 61114409*x^6 + 4548317073*x^7 + 397323349505*x^8 + ...
RELATED TABLE.
The table of coefficients of x^k/k! in exp( n*(2*n+1)*x ) / A(x) begins
  n = 1: [1,  0,  -49,  -3186,  -445203, -109403892, -41045026725, ...];
  n = 2: [1,  7,    0,  -3872,  -546416, -126698400, -45990717440, ...];
  n = 3: [1, 18,  275,      0,  -664875, -160762914, -55439093817, ...];
  n = 4: [1, 33, 1040,  27900,        0, -196031664, -71849477952, ...];
  n = 5: [1, 52, 2655, 129778,  5408749,          0, -87799444565, ...];
  n = 6: [1, 75, 5576, 407664, 28585872, 1710760608,            0, ...];
  ...
illustrating [x^n] exp( n*(2*n+1)*x ) / A(x) = 0 for n > 0.

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

Cf. A304319, A337458.

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

a(n) ~ sqrt(1-w) * 2^(3*n - 1/4) * n^(n - 1/2) / (sqrt(Pi) * exp(n) * (2-w)^n * w^(n + 1/4)), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, May 18 2025

cp's OEIS Frontend

A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

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A337457 O.g.f. A(x) satisfies: [x^n] exp( n*(n-1) * x/A(x) ) = 0 for n > 0.

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A337576 O.g.f. A(x) satisfies: [x^n] exp( n^2*(n+1) * x ) / A(x)^(n+1) = 0 for n > 0.

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A337578 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1)^2 * x ) / A(x)^(n+1) = 0 for n > 0.

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A337579 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1)^2 * x ) / A(x)^((n+1)^2) = 0 for n>0.

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A383559 O.g.f. A(x) satisfies: [x^n] exp( n*(2*n+1)*x ) / A(x) = 0 for n > 0.

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A337457 O.g.f. A(x) satisfies: [x^n] exp( n(n-1) x/A(x) ) = 0 for n > 0.

A337576 O.g.f. A(x) satisfies: [x^n] exp( n^2(n+1) x ) / A(x)^(n+1) = 0 for n > 0.

A337578 O.g.f. A(x) satisfies: [x^n] exp( n(n+1)^2 x ) / A(x)^(n+1) = 0 for n > 0.

A337579 O.g.f. A(x) satisfies: [x^n] exp( n(n+1)^2 x ) / A(x)^((n+1)^2) = 0 for n>0.

A383559 O.g.f. A(x) satisfies: [x^n] exp( n(2n+1)*x ) / A(x) = 0 for n > 0.