A217757 -id:A217757 - 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

A238695 Decimal expansion of Product_{k>=0} (1+1/k!).

Original entry on oeis.org

7, 3, 6, 4, 3, 0, 8, 2, 7, 2, 3, 6, 7, 2, 5, 7, 2, 5, 6, 3, 7, 2, 7, 7, 2, 5, 0, 9, 6, 3, 1, 0, 5, 3, 0, 9, 5, 6, 5, 4, 2, 5, 6, 8, 3, 6, 0, 6, 8, 9, 0, 7, 6, 6, 0, 7, 9, 2, 5, 5, 4, 9, 5, 3, 6, 9, 6, 2, 3, 8, 1, 6, 4, 4, 0, 7, 6, 2, 3, 9, 8, 1, 9, 8, 1, 4, 0, 5, 0, 5, 6, 3, 7, 1, 4, 8, 1, 7, 9, 0, 3, 2, 7, 2, 4, 9, 3, 9, 5, 7, 4, 5, 6, 0, 2, 1

Offset: 1

Frederick Reckless, Mar 03 2014

Conjectured to be irrational, transcendental and normal, none have been shown. Product is sometimes taken from n=1, leading to half the stated value.

			7.3643082723672572563727725096310530956542568360689...

Lucian Craciun, Table of n, a(n) for n = 1..10000
StackExchange, Is Product_{n>=0} (1+1/n!) = e^2?, Apr 06 2013
StackExchange, How to compute Product_{n>=1} (1+1/n!)?, Aug 05 2013
Wolfram Alpha, Product_{n>=0} (1+1/n!)

Cf. A072334, A217757, A282529.

Cf. A387175 (continued fraction).

evalf(product(1+1/n!, n = 0..infinity), 100) # Lucian Craciun, Feb 17 2017

RealDigits[Product[1+1/n!, {n, 0, 75}], 10, 106][[1]] (* Robert G. Wilson v, Mar 19 2014 *)

prodinf(n=1,1+1/gamma(n)) \\ Charles R Greathouse IV, Nov 13 2014

Added more digits from b-file, so as to cover exactly three full rows of text. - Lucian Craciun, Feb 22 2017

A306729 a(n) = Product_{i=0..n, j=0..n} (i! + j!).

Original entry on oeis.org

2, 16, 5184, 9559130112, 109045776752640000000000, 27488263744928988967331390258832998400000000000, 1147897050240877062218236820013018349788772091106840426434074807527014400000000000000

Offset: 0

Vaclav Kotesovec, Mar 06 2019

nonn

Cf. A000178, A055462, A079478, A203482, A217757, A323717, A324403, A325052.

Table[Product[i! + j!, {i, 0, n}, {j, 0, n}], {n, 0, 7}]
Clear[a]; a[n_] := a[n] = If[n == 0, 2, a[n-1] * Product[k! + n!, {k, 0, n}]^2 / (2*n!)]; Table[a[n], {n, 0, 7}] (* Vaclav Kotesovec, Mar 27 2019 *)
Table[Product[Product[k! + j!, {k, 0, j}], {j, 1, n}]^2 / (2^(n-1) * BarnesG[n + 2]), {n, 0, 7}] (* Vaclav Kotesovec, Mar 27 2019 *)

from math import prod, factorial as f
def a(n): return prod(f(i)+f(j) for i in range(n) for j in range(n))
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Feb 16 2021

a(n) ~ c * 2^(n^2/2 + 2*n) * Pi^(n^2/2 + n) * n^(2*n^3/3 + 2*n^2 + 11*n/6 + 5/2) / exp(8*n^3/9 + 2*n^2 + n), where c = A324569 = 62.14398692334529025548974541735...

a(n) = a(n-1) * A323717(n)^2 / (2*n!). - Vaclav Kotesovec, Mar 28 2019

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...

A325050 a(n) = Product_{k=0..n} (k!^2 + 1).

Original entry on oeis.org

2, 4, 20, 740, 426980, 6148938980, 3187616116170980, 80970552724144881738980, 131634021973939424914920841290980, 17333817381151204925617274632152908873802980, 228254990993381085562170532497621436371926846785405002980

Offset: 0

Vaclav Kotesovec, Mar 26 2019

nonn

Cf. A055209, A101686, A217757, A238695, A325048.

Table[Product[k!^2 + 1, {k, 0, n}], {n, 0, 12}]
Table[BarnesG[n+2]^2 * Product[1 + 1/k!^2, {k, 0, n}], {n, 0, 12}]

a(n) ~ c * n^(n^2 + 2*n + 5/6) * (2*Pi)^(n+1) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where c = Product_{k>=0} (1 + 1/k!^2) = 5.1481781945902396880952694880498895... and A is the Glaisher-Kinkelin constant A074962.

A355540 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n} (x - k!) expanded in decreasing powers of x, with row 0 = {1}.

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -10, 29, -32, 12, 1, -34, 269, -728, 780, -288, 1, -154, 4349, -33008, 88140, -93888, 34560, 1, -874, 115229, -3164288, 23853900, -63554688, 67633920, -24883200, 1, -5914, 4520189, -583918448, 15971865420, -120287210688, 320383261440, -340899840000, 125411328000

Offset: 0

Thomas Scheuerle, Jul 06 2022

Essentially the same as A136457 with rows in reversed order.

Let M be an n X n matrix filled by Bell numbers A000110(j+k-2) with rows and columns j = 1..n, k = 1..n; then its determinant equals unsigned T(n, n). If we use A000110(j+k), the determinant will equal unsigned T(n+1, n). Can we find a general formula for T(n+m, n) based on determinants of matrices and Bell numbers?

			The triangle begins:
  1;
  1,   -1;
  1,   -2,      1;
  1,   -4,      5,       -2;
  1,  -10,     29,      -32,       12;
  1,  -34,    269,     -728,      780,      -288;
  1, -154,   4349,   -33008,    88140,    -93888,    34560;
  1, -874, 115229, -3164288, 23853900, -63554688, 67633920, -24883200;
  ...
Row 4: x^4 - 10*x^3 + 29*x^2 - 32*x + 12 = (x-0!)*(x-1!)*(x-2!)*(x-3!).
Illustration of T(1 to 5,1) as tree structure:
.
. o        o         o            o                         o
.          o         o            o                         o
.                   o o          o o                       o o
.                              ooo ooo                   ooo ooo
.                                             oooo oooo oooo oooo oooo oooo
. 1 +1 =   2 +2 =    4 +2*3 =     10 +6*4 =                 34
.
Illustration of T(2 to 4,2) as tree structure:
.
. o         o              -----o-----
.        o     o          o           o
.        o     o       ---o---     ---o---
.                     o   o   o   o   o   o
.                     o   o   o   o   o   o
.                    o o o o o o o o o o o o
. 1 +2*2 =  5 +6*4 =            29
.
Illustration of T(3 to 4,3) as tree structure:
.            ------------
. oo     ---o---      ---o---
.       o   o   o    o   o   o
.      o o o o o o  o o o o o o
.      o o o o o o  o o o o o o
.  2  +6*5 =      32

Cf. A000110, A000178, A000522, A003422, A136457, A203227, A217757.
Cf. A008276 (The Stirling numbers of the first kind in reverse order).
Cf. A039758 (Coefficients for polynomials with roots in odd numbers).
Cf. A349226 (Coefficients for polynomials with roots in x^x).

T(n, k) = polcoeff(prod(m=0, n-1, (x-m!)), n-k);

T(n, 0) = 1.
T(n, 1) = -A003422(n).
T(n, 2) = Sum_{m=0..n-1} !m*m!.
T(n, k) = Sum_{m=0..n-1} -T(m, k-1)*m!.
T(n, n) = (-1)^n*A000178(n).
T(n, n-1) = -(-1)^n*A203227(n), for n > 0.
T(n+1, n) = (-1)^n*A000178(n)*A000522(n).
Sum_{m=0..k} T(n, k) = 0, for n > 0.
Sum_{m=0..k} abs(T(n, k)) = A217757(n+1).

cp's OEIS Frontend

A374848 Obverse convolution A000045**A000045; see Comments.

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A238695 Decimal expansion of Product_{k>=0} (1+1/k!).

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A306729 a(n) = Product_{i=0..n, j=0..n} (i! + j!).

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

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A325050 a(n) = Product_{k=0..n} (k!^2 + 1).

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A355540 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n} (x - k!) expanded in decreasing powers of x, with row 0 = {1}.

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