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 1-10 of 11 results. Next

A002397 a(n) = n! * lcm({1, 2, ..., n+1}).

Original entry on oeis.org

1, 2, 12, 72, 1440, 7200, 302400, 4233600, 101606400, 914457600, 100590336000, 1106493696000, 172613016576000, 2243969215488000, 31415569016832000, 942467070504960000, 256351043177349120000, 4357967734014935040000, 1490424965033107783680000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This term appears in the numerator of several sequences of coefficients used in numerical solutions of ordinary differential equations.

Examples

			5! is 120, and the least common multiple of 2, 3, 4, 5 and 6 is 60, so a(5) = 7200.

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A010796. Row sums of A260780, also of A260781.
The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

Programs

Formula

a(n) = n! * lcm{1,2,...,n+1} = n!*A003418(n+1). - Sean A. Irvine, Nov 07 2013

Extensions

More terms from Sean A. Irvine, Nov 07 2013
More terms from Jack W Grahl, Feb 27 2021

A002398 Coefficients for step-by-step integration.

Original entry on oeis.org

1, 3, 23, 165, 3802, 21385, 993605, 15198435, 394722916, 3814933122, 447827009070, 5229570190845, 862250830559382, 11802457085079375, 173406732097447849, 5443765223302501095, 1545512798280174555832, 27361456077246355572508, 9725198808628092900136884, 191684785790597591594500398
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This is the coefficient of f(x0) in the estimate for y(x1) - y(x0).

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A027486.
Column 0 of A260780.
The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

Extensions

More terms from Jack W Grahl, Feb 28 2021

A002399 Coefficients for step-by-step integration.

Original entry on oeis.org

1, 16, 177, 5548, 39615, 2236440, 40325915, 1207505768, 13229393814, 1737076976040, 22446050738265, 4058838484620084, 60476452041557409, 961082989270516112, 32455938583801467735, 9864953815464307351792, 186195769473110823077652, 70295408103581008790661648, 1466826914074651870368663750
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

These are the negated coefficients of f(x_{-1}) in the estimate for y(x1) - y(x0).

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 1 (negated) of A260780.
The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

Extensions

More terms from Jack W Grahl, Feb 28 2021

A002400 Coefficients for step-by-step integration.

Original entry on oeis.org

5, 111, 5232, 49910, 3527745, 76435695, 2673350008, 33507517680, 4954123399050, 71186377398675, 14169975006172392, 230478985529218998, 3970388091885696481, 144475785096372785055, 47074452451240708494000, 948198128552832829175504, 380523626987174239611912012, 8410876353715824882741160170
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Comments

These are the coefficients of f(x_{-2}) in the estimate for y(x1) - y(x0).

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 2 of A260780.
The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

Extensions

More terms from Jack W Grahl, Feb 28 2021

A002401 Coefficients for step-by-step integration.

Original entry on oeis.org

1, 1, 5, 27, 502, 2375, 95435, 1287965, 29960476, 262426878, 28184365650, 303473091075, 46437880787562, 593196287807409, 8172332906336599, 241563260379065625, 64808657541894257992, 1087738506483388123364, 367580830209839294339148, 6906008426663826491899602, 136666305828261517346022452
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are the coefficients of the n-th forward difference of f in the estimate for y(x1) - y(x0), also the coefficients of f(x0) in the estimate for y(x0) - y(x1).

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 0 of A260781.
The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

Formula

a(n) = lcm{1,2,...,n+1} * Sum_{k=0..n}(1/n+1-k)*s(-(n-1),k,n) where s(l,m,n) are the generalized Stirling numbers of the first kind. - Sean A. Irvine, Nov 10 2013

Extensions

More terms from Sean A. Irvine, Nov 10 2013
More terms from Jack W Grahl, Feb 28 2021

A002402 Coefficients for step-by-step integration.

Original entry on oeis.org

1, 8, 57, 1292, 7135, 325560, 4894715, 125078632, 1190664342, 137798986920, 1587893097945, 258558380321076, 3497709055775649, 50821738502398864, 1578753057237451095, 443765620067972169968, 7782162960545369351956, 2741163034641146307693072, 53564617257321061756508358, 1100369599246721484969558920
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

These are the coefficients of f(x_{-1}) in the estimate for y(x0) - y(x1).

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 1 of A260781.
The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

Extensions

More terms by Jack W Grahl, Feb 28 2021

A002404 Coefficients for step-by-step integration.

Original entry on oeis.org

3, -16, 111, -2548, 14385, -672360, 10351845, -270594968, 2631486186, -310710613080, 3648232023975, -604596371658444, 8315244191734623, -122717408718016112, 3868618892876082345, -1102643727493413977872, 19593301788429800483052, -6988461512994426036295152, 138198195880599649938536250
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are the coefficients of f(x_{-n}) in the estimate for y(x1) - y(x0) which has n + 1 terms.

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

If every other term is negated, this is a diagonal of A260780.
The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

Extensions

More terms from Jack W Grahl, Feb 28 2021

A002405 Coefficients for step-by-step integration.

Original entry on oeis.org

1, -1, -1, -3, -38, -135, -4315, -48125, -950684, -7217406, -682590930, -6554931075, -903921420138, -10496162430897, -132415122967127, -3606726811032345, -896549281211592008, -14008671728814262500, -4425739007479443851340
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

All the terms except the first term are negative. - Sean A. Irvine, Nov 10 2013
a(n) / A002397(n) is the coefficient of the n-th forward difference of f in the estimate of y(x0) - y(x1).

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

With different signs, this is the leading diagonal of A260781.
The coefficients used in numerical integration are given by fractions with A002397 as the denominators.
A002401 is the corresponding sequence for the symmetric method of estimation.
The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

Formula

a(n) = lcm{1,2,...,n+1} * Sum_{k=0..n}((-1)^(n-k)/n+1-k)*s(-(n-1),k,n) where s(l,m,n) are the generalized Stirling numbers of the first kind. - Sean A. Irvine, Nov 10 2013

Extensions

More terms from Sean A. Irvine, Nov 10 2013
More terms from Jack W Grahl, Feb 28 2021

A002406 Coefficients for step-by-step integration.

Original entry on oeis.org

1, 8, -15, 212, -865, 31560, -397285, 8760472, -73512810, 7619823960, -79612742055, 11869626289356, -148201090063455, 2000757995572336, -58073355854498985, 15325818191986269968, -253388757170439526636, 84454267865884467099120, -1566608640281391343515450, 30637801046762651850275960
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are the coefficients of f(x_{-n}) in the estimate for y(x0) - y(x1) which has n + 2 terms.

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Second diagonal of A260781.
The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

Extensions

More terms from Jack W Grahl, Feb 28 2021

A260780 Triangle read by rows: coefficients for predictor y(x_1) for step-by-step integration.

Original entry on oeis.org

1, 3, -1, 23, -16, 5, 165, -177, 111, -27, 3802, -5548, 5232, -2548, 502, 21385, -39615, 49910, -36490, 14385, -2375, 993605, -2235440, 3527745, -3441280, 2035695, -672360, 95435
Offset: 0

Views

Author

N. J. A. Sloane, Aug 02 2015

Keywords

Examples

			Triangle begins:
     1;
     3,    -1;
    23,   -16,    5;
   165,  -177,  111,   -27;
  3802, -5548, 5232, -2548, 502;
  ...

Links

Crossrefs

Row sums = A002397.
Diagonals (up to signs in some cases) include A002398, A002399, A002400, A002401, A002404.
The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

Extensions

a(15)-a(27) from Robert Price, Sep 09 2019
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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