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 1-3 of 3 results.

A337458 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * x ) / A(x)^(n+1) = 0 for n > 0.

Original entry on oeis.org

1, 1, 2, 11, 130, 2450, 63012, 2040779, 79377914, 3594766694, 185457776252, 10725423627006, 686721189003668, 48200778475446916, 3679104677398632520, 303348177377608050219, 26865664102518601306154, 2543352040870175109554654, 256296085507636954980717708, 27390678829206902911266889386
Offset: 0

Views

Author

Paul D. Hanna, Aug 28 2020

Keywords

Comments

It is remarkable that this sequence consists entirely of integers.

Examples

			O.g.f.: A(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 + 185457776252*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+1)*x) / A(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 ) / A(x)^(n+1) = 0 for n>0.
RELATED SERIES.
Define B(x) = A(x/B(x)), which begins
B(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 + ... + A337457(n)*x^n + ...
then the table of coefficients of x^k/k! in exp( n*(n-1)*x/B(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/B(x) ) = 0 for n>0.
Also note that B(x) = x/Series_Reversion( x*A(x) ) and A(x) = B(x*A(x)).

Links

Crossrefs

Cf. A337457, A304319.

Programs

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

Formula

Given o.g.f. A(x), define B(x) = A(x/B(x)), then B(x) is the o.g.f. of A337457 and satisfies [x^n] exp( n*(n-1)*x/B(x) ) = 0 for n>0.
a(n) ~ sqrt(1-c) * 2^(2*n - 3/2) * n^(n - 3/2) / (sqrt(Pi) * c^n * (2-c)^(n-1) * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

A337575 O.g.f. A(x) satisfies: [x^n] exp( n*(n-1)^2 * x/A(x) ) = 0 for n > 0.

Original entry on oeis.org

1, 1, 13, 907, 153145, 46602295, 22140651001, 15084920403375, 13929456839705657, 16740856184792482831, 25396842996449548203625, 47478179622583931337645823, 107267415766722597731672066713, 288206818852524037700531966913487
Offset: 0

Views

Author

Paul D. Hanna, Sep 02 2020

Keywords

Comments

It is remarkable that this sequence consists entirely of integers.

Examples

			O.g.f.: A(x) = 1 + x + 13*x^2 + 907*x^3 + 153145*x^4 + 46602295*x^5 + 22140651001*x^6 + 15084920403375*x^7 + 13929456839705657*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n*(n-1)^2 * x/A(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/A(x) ) = 0 for n>0.
RELATED SERIES.
Define B(x) = A(x*B(x)), which begins
B(x) = 1 + x + 14*x^2 + 947*x^3 + 157190*x^4 + 47437866*x^5 + 22437363324*x^6 + 15246207565643*x^7 + ... + A337576(n)*x^n + ...
then the table of coefficients of x^k/k! in exp( n^2*(n+1)*x ) / B(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 ) / 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)).

Crossrefs

Cf. A337576, A337457, A337577.

Programs

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

Formula

Given o.g.f. A(x), define B(x) = A(x*B(x)), then B(x) is the o.g.f. of A337576 and satisfies [x^n] exp( n^2*(n+1)*x ) / B(x)^(n+1) = 0 for n>0.

A337577 O.g.f. A(x) satisfies: [x^n] exp( n^2*(n-1) * x/A(x) ) = 0 for n > 0.

Original entry on oeis.org

1, 2, 22, 1616, 286700, 90914400, 44673096808, 31286975152640, 29552473932597968, 36189841095064294016, 55768927589536556250016, 105641404186261853184309888, 241363180288689801902138103872, 654744988347389437898766097063424
Offset: 0

Views

Author

Paul D. Hanna, Sep 02 2020

Keywords

Comments

It is remarkable that this sequence consists entirely of integers.

Examples

			O.g.f.: A(x) = 1 + 2*x + 22*x^2 + 1616*x^3 + 286700*x^4 + 90914400*x^5 + 44673096808*x^6 + 31286975152640*x^7 + 29552473932597968*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n*(n-1)^2 * x/A(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/A(x) ) = 0 for n>0.
RELATED SERIES.
Define B(x) = A(x*B(x)), which begins
B(x) = 1 + 2*x + 26*x^2 + 1756*x^3 + 301140*x^4 + 94035272*x^5 + 45829458720*x^6 + 31938032357440*x^7 + ... + A337578(n)*x^n + ...
then the table of coefficients of x^k/k! in exp( n*(n+1)^2*x ) / B(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 ) / 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)).

Crossrefs

Cf. A337578, A337575, A337457.

Programs

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

Formula

Given o.g.f. A(x), define B(x) = A(x*B(x)), then B(x) is the o.g.f. of A337578 and satisfies [x^n] exp( n*(n+1)^2*x ) / B(x)^(n+1) = 0 for n>0.
Showing 1-3 of 3 results.

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