A082480 -id:A082480 - 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

A374654 a(n) = Product_{k=0..n} L(k)+1, where L=A000032 (Lucas numbers).

Original entry on oeis.org

3, 6, 24, 120, 960, 11520, 218880, 6566400, 315187200, 24269414400, 3009407385600, 601881477120000, 194407717109760000, 101480828331294720000, 85649819111612743680000, 116912003087351395123200000, 258141702816871880432025600000

Offset: 0

Clark Kimberling, Jul 25 2024

nonn

Cf. A000032, A000142, A003266, A070825, A082480, A135407, A374655.

w[n_] := Product[LucasL[k] + 1, {k, 0, n}]
Table[w[n], {n, 0, 20}]

a(n) = Product_{k=0..n} L(k)+1, where L=A000032 (Lucas numbers).

A217757 Product_{i=0..n} (i! + 1).

Original entry on oeis.org

2, 4, 12, 84, 2100, 254100, 183206100, 923541950100, 37238134969982100, 13513011656042074430100, 49036030210457135734021310100, 1957361459740805606124917565020990100, 937579272951542930363610919638075856505150100

Offset: 0

Jon Perry, Mar 23 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..42

Cf. A000142, A000178, A054640, A082480, A258325, A260231, A306729.

function factorial(n) {
var i,c=1;
for (i=2;i<=n;i++) c*=i;
return c;
}
a=2;
for (j=1;j<10;j++) {
a*=(factorial(j)+1);
document.write(a+", ");
}

a:= proc(n) a(n):= `if`(n=0, 2, a(n-1)*(n!+1)) end:
seq(a(n), n=0..14);  # Alois P. Heinz, May 20 2013

Table[Product[i!+1,{i,0,n}],{n,0,12}]  (* Geoffrey Critzer, May 04 2013 *)
Rest[FoldList[Times,1,Range[0,15]!+1]] (* Harvey P. Dale, May 28 2013 *)

a(n) ~ c * A000178(n), where c = A238695 = Product_{k>=0} (1 + 1/k!) = 7.364308272367257256372772509631... . - Vaclav Kotesovec, Jul 20 2015

A158472 Triangle read by rows: n-th row is the expansion of the polynomial (x-F1)(x-F2)(x-F3)...(x-Fn).

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -7, 17, -17, 6, 1, -12, 52, -102, 91, -30, 1, -20, 148, -518, 907, -758, 240, 1, -33, 408, -2442, 7641, -12549, 10094, -3120, 1, -54, 1101, -11010, 58923, -173010, 273623, -215094, 65520

Offset: 0

Gary W. Adamson, Mar 20 2009

Row sums of the unsigned triangle = A082480: (1, 2, 4, 12, 48, 288, 2592, ...).

Right border starting with row 1 (unsigned) = A003266: (1, 1, 2, 6, 30, 240, ...).

			First few rows of the unsigned triangle:
  1;
  1,  1;
  1,  2,    1;
  1,  4,    5,     2;
  1,  7,   17,    17,     6;
  1, 12,   52,   102,    91,     30;
  1, 20,  148,   518,   907,    758,    240;
  1, 33,  408,  2442,  7641,  12549,  10094,   3120;
  1, 54, 1101, 11010, 58923, 173010, 273623, 215094, 65520;
  ...
Example: row 5 is x^5 - 12x^4 + 52x^3 - 102x^2 + 91x - 30
= (x-1)*(x-1)*(x-2)*(x-3)*(x-5).

Alois P. Heinz, Rows n = 0..98, flattened

Cf. A000045, A082480, A003266.

p:= proc(n) option remember; expand(`if`(n=0, 1,
      p(n-1)*(x-(<<0|1>, <1|1>>^n)[1, 2])))
    end:
T:= (n, k)-> coeff(p(n), x, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Nov 06 2016

Array[Reverse@ CoefficientList[Times @@ Array[(x - Fibonacci@ #) &, #], x] &, 9, 0] // Flatten (* Michael De Vlieger, Apr 21 2019 *)

row(n) = Vec(prod(k=1, n, x-fibonacci(k)));
for (n=0, 10, print(row(n))); \\ Michel Marcus, Apr 22 2019

One term corrected by Alois P. Heinz, Nov 06 2016

A258325 a(n) = Product_{k=1..n} (1 + p(k)), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 2, 6, 24, 144, 1152, 13824, 221184, 5087232, 157704192, 6781280256, 386532974592, 30149572018176, 3075256345853952, 418234863036137472, 74027570757396332544, 17174396415715949150208, 5117970131883352846761984, 1975536470906974198850125824

Offset: 0

Vaclav Kotesovec, Jul 19 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150
Vaclav Kotesovec, The partition factorial constant and asymptotics of the sequence A058694

Cf. A000041, A058694, A078506, A082480.

a:= proc(n) option remember: `if`(n<1, 1,
      (1+combinat[numbpart](n))*a(n-1))
    end:
seq(a(n), n=0..20);

Table[Product[PartitionsP[k]+1,{k,1,n}],{n,0,20}]

a(n) ~ c * A058694(n), where c = Product_{k>=1} (1 + 1/p(k)) = 7.60150293364724365227288154074110141857580676049277152624021470033199348...

a(0)=1 prepended by Alois P. Heinz, Jul 26 2015

A260231 a(n) = Product_{k=1..n} (1 + k^k).

Original entry on oeis.org

2, 10, 280, 71960, 224946960, 10495350312720, 8643382777938679680, 145011908479540041684850560, 56180584638978557924165229531974400, 561805846445966163880630853243909229531974400, 160289764609087349005207761687490741791453382934816332800

Offset: 1

Vaclav Kotesovec, Jul 20 2015

nonn

Cf. A000312, A002109, A028361, A054640, A082480, A217757, A258325.

Table[Product[1+k^k,{k,1,n}],{n,1,12}]
FoldList[Times,Table[1+k^k,{k,12}]] (* Harvey P. Dale, Jul 19 2025 *)

a(n) ~ c * A002109(n), where c = Product_{k>=1} (1 + 1/k^k) = 2.60361190459951423330221282635022049352582879064202503882732200701325334...

A374662 a(n) = (1/2)*Product_{k=0..n} (F(k)+2), where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 3, 9, 36, 180, 1260, 12600, 189000, 4347000, 156492000, 8920044000, 811724004000, 118511704584000, 27850250577240000, 10555244968773960000, 6459809920889663520000, 6388752011759877221280000, 10215614466804043676826720000, 26417579011155256948273897920000

Offset: 0

Clark Kimberling, Aug 03 2024

nonn

a(n+1)/a(n) is an integer for n>=0, so (a(n)) is a divisibility sequence.

Cf. A000045, A082480.

q[n_] := Fibonacci[n]
p[n_] := Product[q[k] + 2, {k, 0, n}]
Table[Simplify[p[n]/2], {n, 0, 20}]

a(n) = prod(k=0, n, fibonacci(k)+2)/2; \\ Michel Marcus, Aug 04 2024

A374982 a(n) = (1/3)*Product_{k=0..n} (F(k)+3), where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 4, 16, 80, 480, 3840, 42240, 675840, 16220160, 600145920, 34808463360, 3202378629120, 470749658480640, 111096919401431040, 42216829372543795200, 25878916405369346457600, 25620127241315652993024000, 40992203586105044788838400000

Offset: 0

Clark Kimberling, Aug 03 2024

nonn

Trivially, a(n+1)/a(n) is an integer for n>=0, so (a(n)) is a divisibility sequence.

Cf. A000045, A082480.

q[n_] := Fibonacci[n]
p[n_] := Product[q[k] + 3, {k, 0, n}]
Table[(1/3)*Simplify[p[n]], {n, 0, 20}]

a(n) = prod(k=0, n, fibonacci(k)+3)/3; \\ Michel Marcus, Aug 04 2024

cp's OEIS Frontend

A374848 Obverse convolution A000045**A000045; see Comments.

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A374654 a(n) = Product_{k=0..n} L(k)+1, where L=A000032 (Lucas numbers).

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A217757 Product_{i=0..n} (i! + 1).

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A158472 Triangle read by rows: n-th row is the expansion of the polynomial (x-F1)*(x-F2)*(x-F3)*...*(x-Fn).

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A258325 a(n) = Product_{k=1..n} (1 + p(k)), where p(k) is the partition function A000041.

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A260231 a(n) = Product_{k=1..n} (1 + k^k).

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A374662 a(n) = (1/2)*Product_{k=0..n} (F(k)+2), where F=A000045 (Fibonacci numbers).

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A374982 a(n) = (1/3)*Product_{k=0..n} (F(k)+3), where F=A000045 (Fibonacci numbers).

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A158472 Triangle read by rows: n-th row is the expansion of the polynomial (x-F1)(x-F2)(x-F3)...(x-Fn).