A002398 -id:A002398 - cp's OEIS frontend

A374848 Obverse convolution A000045**A000045; see Comments.

0, 1, 2, 16, 162, 3600, 147456, 12320100, 2058386904, 701841817600, 488286500625000, 696425232679321600, 2038348954317776486400, 12259459134020160144810000, 151596002479762016373851690400, 3855806813438155578522841251840000

Offset: 0

Clark Kimberling, Jul 31 2024

nonn

The obverse convolution of sequences
  s = (s(0), s(1), ...) and t = (t(0), t(1), ...)
is introduced here as the sequence s**t given by
s**t(n) = (s(0)+t(n)) * (s(1)+t(n-1)) * ... * (s(n)+t(0)).
Swapping * and + in the representation s(0)*t(n) + s(1)*t(n-1) + ... + s(n)*t(0)
of ordinary convolution yields s**t.
If x is an indeterminate or real (or complex) variable, then for every sequence t of real (or complex) numbers, s**t is a sequence of polynomials p(n) in x, and the zeros of p(n) are the numbers -t(0), -t(1), ..., -t(n).
Following are abbreviations in the guide below for triples (s, t, s**t):
F = (0,1,1,2,3,5,...) = A000045, Fibonacci numbers
L = (2,1,3,4,7,11,...) = A000032, Lucas numbers
P = (2,3,5,7,11,...) = A000040, primes
T = (1,3,6,10,15,...) = A000217, triangular numbers
C = (1,2,6,20,70, ...) = A000984, central binomial coefficients
LW = (1,3,4,6,8,9,...) = A000201, lower Wythoff sequence
UW = (2,5,7,10,13,...) = A001950, upper Wythoff sequence
[ ] = floor
In the guide below, sequences s**t are identified with index numbers Axxxxxx; in some cases, s**t and Axxxxxx differ in one or two initial terms.
Table 1. s = A000012 = (1,1,1,1...) = (1);
t = A000012; 1                 s**t = A000079; 2^(n+1)
t = A000027; n                 s**t = A000142; (n+1)!
t = A000040, P                 s**t = A054640
t = A000040, P           (1/3) s**t = A374852
t = A000079, 2^n               s**t = A028361
t = A000079, 2^n         (1/3) s**t = A028362
t = A000045, F                 s**t = A082480
t = A000032, L                 s**t = A374890
t = A000201, LW                s**t = A374860
t = A001950, UW                s**t = A374864
t = A005408, 2*n+1             s**t = A000165, 2^n*n!
t = A016777, 3*n+1             s**t = A008544
t = A016789, 3*n+2             s**t = A032031
t = A000142, n!                s**t = A217757
t = A000051, 2^n+1             s**t = A139486
t = A000225, 2^n-1             s**t = A006125
t = A032766, [3*n/2]           s**t = A111394
t = A034472, 3^n+1             s**t = A153280
t = A024023, 3^n-1             s**t = A047656
t = A000217, T                 s**t = A128814
t = A000984, C                 s**t = A374891
t = A279019, n^2-n             s**t = A130032
t = A004526, 1+[n/2]           s**t = A010551
t = A002264, 1+[n/3]           s**t = A264557
t = A002265, 1+[n/4]           s**t = A264635
Sequences (c)**L, for c=2..4: A374656 to A374661
Sequences (c)**F, for c=2..6: A374662, A374662, A374982 to A374855
The obverse convolutions listed in Table 1 are, trivially, divisibility sequences. Likewise, if s = (-1,-1,-1,...) instead of s = (1,1,1,...), then s**t is a divisibility sequence for every choice of t; e.g. if s = (-1,-1,-1,...) and t = A279019, then s**t = A130031.
Table 2. s = A000027 = (0,1,2,3,4,5,...) = (n);
t = A000027, n                 s**t = A007778, n^(n+1)
t = A000290, n^2               s**t = A374881
t = A000040, P                 s**t = A374853
t = A000045, F                 s**t = A374857
t = A000032, L                 s**t = A374858
t = A000079, 2^n               s**t = A374859
t = A000201, LW                s**t = A374861
t = A005408, 2*n+1             s**t = A000407, (2*n+1)! / n!
t = A016777, 3*n+1             s**t = A113551
t = A016789, 3*n+2             s**t = A374866
t = A000142, n!                s**t = A374871
t = A032766, [3*n/2]           s**t = A374879
t = A000217, T                 s**t = A374892
t = A000984, C                 s**t = A374893
t = A038608, n*(-1)^n          s**t = A374894
Table 3. s = A000290 = (0,1,4,9,16,...) = (n^2);
t = A000290, n^2               s**t = A323540
t = A002522, n^2+1             s**t = A374884
t = A000217, T                 s**t = A374885
t = A000578, n^3               s**t = A374886
t = A000079, 2^n               s**t = A374887
t = A000225, 2^n-1             s**t = A374888
t = A005408, 2*n+1             s**t = A374889
t = A000045, F                 s**t = A374890
Table 4. s = t;
s = t = A000012, 1             s**s = A000079; 2^(n+1)
s = t = A000027, n             s**s = A007778, n^(n+1)
s = t = A000290, n^2           s**s = A323540
s = t = A000045, F             s**s = this sequence
s = t = A000032, L             s**s = A374850
s = t = A000079, 2^n           s**s = A369673
s = t = A000244, 3^n           s**s = A369674
s = t = A000040, P             s**s = A374851
s = t = A000201, LW            s**s = A374862
s = t = A005408, 2*n+1         s**s = A062971
s = t = A016777, 3*n+1         s**s = A374877
s = t = A016789, 3*n+2         s**s = A374878
s = t = A032766, [3*n/2]       s**s = A374880
s = t = A000217, T             s**s = A375050
s = t = A005563, n^2-1         s**s = A375051
s = t = A279019, n^2-n         s**s = A375056
s = t = A002398, n^2+n         s**s = A375058
s = t = A002061, n^2+n+1       s**s = A375059
If n = 2*k+1, then s**s(n) is a square; specifically,
s**s(n) = ((s(0)+s(n))*(s(1)+s(n-1))*...*(s(k)+s(k+1)))^2.
If n = 2*k, then s**s(n) has the form 2*s(k)*m^2, where m is an integer.
Table 5. Others
s = A000201, LW      t = A001950, UW         s**t = A374863
s = A000045, F       t = A000032, L          s**t = A374865
s = A005843, 2*n     t = A005408, 2*n+1      s**t = A085528, (2*n+1)^(n+1)
s = A016777, 3*n+1   t = A016789, 3*n+2      s**t = A091482
s = A005408, 2*n+1   t = A000045, F          s**t = A374867
s = A005408, 2*n+1   t = A000032, L          s**t = A374868
s = A005408, 2*n+1   t = A000079, 2^n        s**t = A374869
s = A000027, n       t = A000142, n!         s**t = A374871
s = A005408, 2*n+1   t = A000142, n!         s**t = A374872
s = A000079, 2^n     t = A000142, n!         s**t = A374874
s = A000142, n!      t = A000045, F          s**t = A374875
s = A000142, n!      t = A000032, L          s**t = A374876
s = A005408, 2*n+1   t = A016777, 3*n+1      s**t = A352601
s = A005408, 2*n+1   t = A016789, 3*n+2      s**t = A064352
Table 6. Arrays of coefficients of s(x)**t(x), where s(x) and t(x) are polynomials
s(x)              t(x)            s(x)**t(x)
 n                 x              A132393
n^2                x              A269944
x+1               x+1             A038220
x+2               x+2             A038244
 x                x+3             A038220
nx                x+1             A094638
 1              x^2+x+1           A336996
n^2 x             x+1             A375041
n^2 x            2x+1             A375042
n^2 x             x+2             A375043
2^n x             x+1             A375044
2^n              2x+1             A375045
2^n              x+2              A375046
x+1              F(n)             A375047
x+1             x+F(n)            A375048
x+F(n)          x+F(n)            A375049

			a(0) = 0 + 0 = 0
a(1) = (0+1) * (1+0) = 1
a(2) = (0+1) * (1+1) * (1+0) = 2
a(3) = (0+2) * (1+1) * (1+1) * (2+0) = 16
As noted above, a(2*k+1) is a square for k>=0. The first 5 squares are 1, 16, 3600, 12320100, 701841817600, with corresponding square roots 1, 4, 60, 3510, 837760.
If n = 2*k, then s**s(n) has the form 2*F(k)*m^2, where m is an integer and F(k) is the k-th Fibonacci number; e.g., a(6) = 2*F(3)*(192)^2.

Cf. A000045, A374850-to-A374881, A374884-to-A374894, A375041-to-A375049.

a:= n-> (F-> mul(F(n-j)+F(j), j=0..n))(combinat[fibonacci]):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 02 2024

s[n_] := Fibonacci[n]; t[n_] := Fibonacci[n];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]

a(n)=prod(k=0, n, fibonacci(k) + fibonacci(n-k)) \\ Andrew Howroyd, Jul 31 2024

a(n) ~ c * phi^(3*n^2/4 + n) / 5^((n+1)/2), where c = QPochhammer(-1, 1/phi^2)^2/2 if n is even and c = phi^(1/4) * QPochhammer(-phi, 1/phi^2)^2 / (phi + 1)^2 if n is odd, and phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 01 2024

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

N. J. A. Sloane

nonn

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

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

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).

Jack W Grahl, Table of n, a(n) for n = 0..100
Jack W Grahl, Explanation of the use of this sequence.
Jack W Grahl, Python code to calculate this and related sequences.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

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.

a(n) = n!*lcm([1..n+1]); \\ Michel Marcus, Oct 15 2023

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

More terms from Sean A. Irvine, Nov 07 2013

More terms from Jack W Grahl, Feb 27 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

N. J. A. Sloane

nonn

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

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).

Jack W Grahl, Table of n, a(n) for n = 1..100
Jack W Grahl, Explanation of how the sequence was calculated.
Jack W Grahl, Python code to calculate this and related sequences.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

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.

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

N. J. A. Sloane

nonn

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

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).

Jack W Grahl, Table of n, a(n) for n = 2..100
Jack W Grahl, Explanation of how this sequence is calculated.
Jack W Grahl, Python code to calculate this and related sequences.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

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.

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

N. J. A. Sloane

nonn

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).

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).

Jack W Grahl, Table of n, a(n) for n = 0..100
Jack W Grahl, Explanation of how this sequence is calculated
Jack W Grahl, Python code to calculate this and related sequences
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

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.

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

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

N. J. A. Sloane

nonn

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

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).

Jack W Grahl, Table of n, a(n) for n = 1..100
Jack W Grahl, Explanation of how this sequence is calculated.
Jack W Grahl, Python code to calculate this and related sequences.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

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.

More terms by Jack W Grahl, Feb 28 2021

A002403 Coefficients for step-by-step integration.

Original entry on oeis.org

1, 15, 528, 3990, 232305, 4262895, 128928632, 1420184304, 186936865290, 2416826727315, 436683783190248, 6495589851083190, 102988034105173217, 3468347338313592735, 1050976389766688264880, 19771777981152440202960, 7439086137698489458667340, 154685313008524836907739370, 3369940174123349111629009120

Offset: 2

N. J. A. Sloane

nonn

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

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).

Jack W Grahl, Table of n, a(n) for n = 2..100
Jack W Grahl, Explanation of how this sequence is calculated.
Jack W Grahl, Python code to calculate this and related sequences.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

Column 2 (negated) 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.

More terms from 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

N. J. A. Sloane

sign

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

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).

Jack W Grahl, Table of n, a(n) for n = 0..100
Jack W Grahl, Explanation of how this sequence is calculated.
Jack W Grahl, Python code to calculate this and related sequences.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

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.

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

N. J. A. Sloane

sign

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).

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).

Jack W Grahl, Table of n, a(n) for n = 0..100
Jack W Grahl, Explanation of how the sequence was calculated
Jack W Grahl, Python code to calculate this and related sequences
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

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.

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

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

N. J. A. Sloane

sign

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

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).

Jack W Grahl, Table of n, a(n) for n = 0..100
Jack W Grahl, Explanation of how this sequence is calculated.
Jack W Grahl, Python code to calculate this and related sequences.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

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.

More terms from Jack W Grahl, Feb 28 2021

cp's OEIS Frontend

A374848 Obverse convolution A000045**A000045; see Comments.

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

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A002399 Coefficients for step-by-step integration.

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A002400 Coefficients for step-by-step integration.

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A002401 Coefficients for step-by-step integration.

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A002402 Coefficients for step-by-step integration.

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A002403 Coefficients for step-by-step integration.

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A002404 Coefficients for step-by-step integration.

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