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 1-2 of 2 results.

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

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 + 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)).

Crossrefs

Cf. A337575, A337458, A337578.

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 +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 A337575 and satisfies [x^n] exp( n*(n-1)^2 * x/B(x) ) = 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 1-2 of 2 results.

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