A137852 -id:A137852 - cp's OEIS frontend

A006973 Dimensions of representations by Witt vectors.

0, 1, 2, 9, 24, 130, 720, 8505, 35840, 412776, 3628800, 42030450, 479001600, 7019298000, 82614884352, 1886805545625, 20922789888000, 374426276224000, 6402373705728000, 134987215801622184, 2379913632645120000

Offset: 1

Simon Plouffe

Starting (1, 2, 9, 24, ...) = row sums of triangle A156792. - Gary W. Adamson, Feb 15 2009

			G.f.: exp(-x)/(1-x) = (1 + 0*x)*(1 + 1*x^2/2!)*(1 + 2*x^3/3!)*(1 + 9*x^4/4!)*
(1 + 24*x^5/5!)*(1 + 130*x^6/6!)*...*(1 + a(n)*x^n/n!)*...
Recurrence: a(7) = -1 - (7*a(1)*a(6) + 21*a(2)*a(5) + 35 a(3)*a(4) + 105*a(1)*a(2)*a(4)) = -1 -(-910 + 504 + 630 - 945) = 720 = 6!. For the recurrence one has to use a(1)=-1. - _Wolfdieter Lang_, Feb 24 2009
G.f. = x^2 + 2*x^3 + 9*x^4 + 24*x^5 + 130*x^6 + 720*x^7 + 8505*x^8 + ...

Reutenauer, Christophe; Sur des fonctions symétriques liées aux vecteurs de Witt et à l'algèbre de Lie libre, Report 177, Dept. Mathématiques et d'Informatique, Univ. Québec à Montréal, Mar 26 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
J. Borwein, Letter to C. Reutenauer, n.d.
Jonathan Borwein and Shi Tuo Lou, Asymptotics of a sequence of Witt vectors, J. Approx. Theory 69 (1992), no. 3, 326-337. Math. Rev. 93f:05007.
Johann Cigler, Some remarks on the power product expansion of the q-exponential series, arXiv:2006.06242 [math.CO], 2020.
Gottfried Helms, A dream of a (number-) sequence, 2007-2009.
C. Reutenauer, Sur des fonctions symétriques liées aux vecteurs de Witt et à l'algèbre de Lie libre, Report 177, Dept. Mathématiques et d'Informatique, Univ. Québec à Montréal, Mar 26 1992. [Annotated scanned copy]
Christophe Reutenauer, Sur des fonctions symétriques reliées aux vecteurs de Witt, [ On symmetric functions related to Witt vectors ] C. R. Acad. Sci. Paris Ser. I Math. 312 (1991), no. 7, 487-490.
C. Reutenauer, Sur des fonctions symétriques reliées aux vecteurs de Witt, [ On symmetric functions related to Witt vectors ] , C. R. Acad. Sci. Paris Ser. I Math. 312 (1991), no. 7, 487-490. (Annotated scanned copy)

Cf. A137852, A156792.

a[n_] := a[n] = If[n < 4, Max[n-1, 0], (n-1)!*(1 + Sum[ k*(-a[k]/k!)^(n/k), {k, Most[Divisors[n]]}])]; Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Jul 19 2012, after 1st PARI program *)
a[ n_]:= If[n<2, 0, a[n] = n! SeriesCoefficient[ Exp[-x]/((1-x) Product[ 1 + a[k] x^k/k!, {k, 2, n-1}]), {x, 0, n}]]; (* Michael Somos, Feb 23 2015 *)

a(n)=if(n<4,max(n-1,0),(n-1)!*(1+sumdiv(n,k, if(k

/* As coefficients in product g.f.: */ a(n)=if(n<2,0,n!*polcoeff((exp(-x+x*O(x^n))/(1-x))/prod(k=0,n-1,1+a(k)*x^k/k! +x*O(x^n)),n)) \\ Paul D. Hanna, Feb 14 2008

G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = exp(-x)/(1-x). - Paul D. Hanna, Feb 14 2008

A recurrence. With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions (fp)) and the multinomial numbers M1(fp(n,m)) (given as M_1 array for any partition in A036038): a(n) = (-1)^n - Sum_{m=2..maxm(n)} ( Sum_{fp from FP(n,m)} (M1(fp)*Product_{j=1..m} ( a(k[j]) ) ), with maxm(n) = A003056(n) = floor((sqrt(1+8*n) -1)/2) and the distinct parts k[j], j=1..m, of the partition of n, n>=2, with input a(1)=-1 (but only for this recurrence). Note that a(1)=0. Proof by comparing coefficients of (x^n)/n! in exp(-x) = (1-x)*Product_{j>=1} ( 1 + a(j)*(x^j)/j! ). See array A008289(n,m) for the cardinality of the set FP(n,m). Another recurrence has been given in the first PARI program line below. - Wolfdieter Lang, Feb 24 2009

More terms from Michael Somos, Oct 07 2001

Further terms from Paul D. Hanna, Feb 14 2008

A147542 Product(1 + a(n)x^n, n=1..infinity) = sum(F(k+1)x^k, k=1..infinity) = 1/(1-x-x^2), where F(n) = A000045(n) (Fibonacci numbers).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 4, 18, 8, 8, 18, 17, 40, 50, 88, 396, 210, 296, 492, 690, 1144, 1776, 2786, 3545, 6704, 10610, 16096, 25524, 39650, 63544, 97108, 269154, 236880, 389400, 589298, 956000, 1459960, 2393538, 3604880, 5739132, 9030450, 14777200

Offset: 1

Neil Fernandez, Nov 06 2008

nonn

A formal infinite product representation for the Fibonacci numbers (A000045(n+1)).

For references see A147541. [R. J. Mathar, Mar 12 2009]

Jean-François Alcover, Table of n, a(n) for n = 1..200
Wolfdieter Lang Two recurrences for the general problem.
R. J. Mathar, Re: polynomial-to-product transform, Maple code (2008). [From _R. J. Mathar_, Mar 12 2009]

Cf. A000045, A137852, A006973, A157159.

m = 200;
sol = Thread[CoefficientList[Sum[Log[1 + a[n] x^n], {n, 1, m}] - Log[1/(1 - x - x^2)] + O[x]^(m + 1), x] == 0] // Solve // First;
Array[a, m] /. sol (* Jean-François Alcover, Oct 22 2019 *)

From Wolfdieter Lang, Mar 06 2009: (Start)
Recurrence I: With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions (fp)):
a(n)= F(n+1) - sum(sum(product(a(k[j]),j=1..m),fp from FP(n,m)),m=2..maxm(n)), with maxm(n):=A003056(n) and the distinct parts k[j], j=1,...,m, of the partition fp of n, n>=3. Inputs a(1)=F(2)=1, a(2)=F(3)=2. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: With the definition of FP(n,m) from the above recurrence I, P(n,m) the general set of partitions of n with m parts, and the multinomial numbers M_0 (given for every partition under A048996):
a(n) = sum((d/n)*(-a(d)^(n/d)),d|n with 1=2; a(1)=F(2)=1. The exponents e(j)>=0 satisfy sum(j*e(j),j=1..n)=n and sum(e(j),j=1..m). The M_0 numbers are m!/product(e(j)!,j=1..n).
Example of recurrence I: a(4) = F(5) - a(1)*a(3) = 5 - 1*1 = 4.
Example of recurrence II: a(4)= 2*(-1)^2 + (1*F(5)-(1/2)*(2*F(2)*F(4) + 1*F(3)^2) + (1/3)*3*F(2)^2*F(3)) = 4. (End)

More terms and revised description from Wolfdieter Lang Mar 06 2009
Edited by N. J. A. Sloane, Mar 11 2009 at the suggestion of Vladeta Jovovic
More terms from R. J. Mathar, Mar 12 2009

A157159 Infinite product representation of series 1 - log(1-x) = 1 + Sum_{j>=1} (j-1)!*(x^j)/j!.

Original entry on oeis.org

1, 1, -1, 10, -16, 126, -526, 10312, -30024, 453840, -2805408, 45779328, -374664720, 7932770496, -67692115440, 2432120198016, -16610113920768, 437275706750208, -5110200130727808, 159305381515284480, -1931470594025607936, 63854116254680514048

Offset: 1

Wolfdieter Lang Mar 06 2009

sign

			Recurrence I: a(7) = 6! - (7*a(1)*a(6) + 21*a(2)*a(5) + 35*a(3)*a(4) + 105*a(1)*a(2)*a(4)) = 720 - (7*126 + 21*(-16) + 35*(-1)*10 + 105*10) = -526.
Recurrence II: a(4) = 3!*(1+2*(-1/2!)^2) + 1 = +10.

Alois P. Heinz, Table of n, a(n) for n = 1..170

Cf. A137852, A006973, A147542.

a:= proc(n) option remember; `if`(n=1, 1, (n-1)!*((-1)^n+add(d*
      (-a(d)/d!)^(n/d), d=numtheory[divisors](n) minus {1, n}))
       +(-1)^(n+1)*add((k-1)!*Stirling1(n, k), k=1..n))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 14 2012

a[n_] := a[n] = If[n == 1, 1, (n-1)!*((-1)^n+Sum[d*(-a[d]/d!)^(n/d), {d, Divisors[n][[2 ;; -2]]}])+(-1)^(n+1)*Sum[(k-1)!*StirlingS1[n, k], {k, 1, n}]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

Definition of a(n): 1-log(1-x) = product(1+a(n)*(x^n)/n!, n=1..infinity) (formal series and product).
Recurrence I. With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions (fp)) and the multinomial numbers M1(fp(n,m)) (given as array in A036038 for any partition) for fp(n,m) from FP(n,m): a(n) = (n-1)! - sum(sum(M1(fp)*product(a(k[j]),j=1..m),fp from FP(n,m)), m=2..maxm(n)), with maxm(n):=A003056(n) and the distinct parts k[j], j=1,...,m, of the partition fp of n, n>=3. Inputs a(1)=1, a(2)=1. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: a(n) = (n-1)!*((-1)^n + sum(d*(-a(d)/d!)^(n/d),d|n with 1A089064(n), n>=2, a(1)=1. A089064(n)=sum(((-1)^(m-1))*(m-1)!)*|S1(n,m)|, m=1..n) with the unsigned Stirling numbers of the first kind |A008275|. See the W. Lang link under A147542 for these recurrences.

More terms from Alois P. Heinz, Aug 14 2012

A353607 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + sin(x).

Original entry on oeis.org

1, 0, -1, 4, -19, 114, -659, 5328, -38375, 400430, -3578279, 44920360, -476298835, 6949878740, -85215100151, 1492480745728, -20903398375855, 382829285287446, -6399968826052559, 136747967762351544, -2394435177245209195, 55602194767215266060, -1123931378903214542099

Offset: 1

Ilya Gutkovskiy, May 07 2022

sign

Cf. A006973, A137852, A170914, A170915, A353608, A353609, A353610, A353611.

nn = 23; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Sin[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A353608 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + sinh(x).

Original entry on oeis.org

1, 0, 1, -4, 21, -126, 1023, -8240, 84745, -864370, 10925883, -133566808, 1994183205, -28455880012, 489891177051, -8112780640000, 158096182329585, -2911196026492074, 64115697136312563, -1328879415116924744, 31920276313015362525, -728711636884140292372

Offset: 1

Ilya Gutkovskiy, May 07 2022

sign

Cf. A006973, A137852, A353607, A353609, A353610, A353611.

nn = 22; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Sinh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A353611 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + tan(x).

Original entry on oeis.org

1, 0, 2, -8, 56, -336, 3184, -27264, 309760, -3297280, 48104704, -624745472, 10591523840, -159594803200, 3133776259072, -56224864108544, 1249919350046720, -24600643845095424, 624022403933077504, -14094091678163140608, 381632216575339397120, -9516741266133420605440

Offset: 1

Ilya Gutkovskiy, May 07 2022

sign

Cf. A000182, A006973, A009006, A137852, A353583, A353584, A353607, A353608, A353609, A353610.

nn = 22; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Tan[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A157164 1/product(1 - a(n)*(x^n)/n!, n=1..infinity) = 1 + sum((x^k)/k, k=1..infinity) = 1 - log(1-x).

Original entry on oeis.org

1, -1, -1, -8, -16, -74, -526, -6768, -30024, -291072, -2805408, -29134896, -374664720, -5276228736, -67692115440, -1404936248064, -16610113920768, -301439595923712, -5110200130727808, -103584959322338304

Offset: 1

Wolfdieter Lang Aug 10 2009

			Recurrence I: a(4) = 6 - 4*a(1)*a(3) - 6*a(2)^2 -12*a(1)^2*a(2) - 24*a(1)^4 = 6-4*(-1)-6-12*(-1) = -8.
Recurrence II: a(4) = -6*(1+1+ 2*(-1/2)^2) + 0!*6 - 1*(8+3) + 2*6 = -8.
Recurrence II (rewritten): a(4) = -6*(1+2*(-1/2)^2) +1 = -8.

W. Lang Recurrences for the general problem.

A137852 (with different signs) for exp(x) = 1/product(1 - a(n)*(x^n)/n!, n=1..infinity).

Recurrence I: With P(n,m) the set of partitions of n with m parts:
a(n)= (n-1)! - sum(sum(M1(n;vec(e))*product(a(j)^e(j),j=1..m), p=(1^e(1),...,n^e(n)) from P(n,m)), m=2..n) for n>=2, with sum(j*e(j),j=1..n)=n, sum(e(j),j=1..n)=m for the partition p of n with m parts. Input a(1)=0!=1. See the array A008284(n,m) for the cardinalities of the sets P(n,m). The M1 numbers are given, for any partition p=(1^e(1),...,n^e(n)) from P(n,m) by M1(n;vec(e)):=n/product(j!^e(j), j=1..n). See the array A036038.
Recurrence II: a(n)= -(n-1)!*((1+(-1)^n) + sum(d*(a(d)/d!)^(n/d),d|n with 1=2; a(1)=0!=1. The set P(n,m) has been defined in recurrence I above. M2(n;vec(e):=n!/product(e(j)!, j=1..n).
Recurrence II (rewritten, due to email from V. Jovovic, Mar 10 2009. See the link for the general derivation):
a(n) = -(n-1)!*sum(d*(a(d)/d!)^(n/d),d|n with 1<=dA089064(n)=[1,0,1,1,8,26,194,1142,...].

A353609 Product_{n>=1} (1 + a(n)x^(2n)/(2*n)!) = cosh(x).

Original entry on oeis.org

1, 1, -14, 393, -14744, 972610, -74928944, 9322093753, -1163849271296, 228519734620776, -44942000161435904, 12717856972091286642, -3539995034294896016384, 1371560847857743301790928, -510461123036204706738612224, 268938575250382935485761673113

Offset: 1

Ilya Gutkovskiy, May 07 2022

sign

Cf. A006973, A137852, A170912, A170913, A353607, A353608, A353610, A353611.

nn = 16; f[x_] := Product[(1 + a[n] x^(2 n)/(2 n)!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Cosh[x], {x, 0, 2 nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A348205 Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + log(1 + x).

Original entry on oeis.org

1, -3, 5, -68, 204, -1394, 16862, -413776, 2377512, -35594832, 558727872, -8067263280, 185546362416, -4108304962176, 82441247589360, -3519099528152064, 50908186083448320, -1465023121035418368, 38998680958184088960, -1219845314470474404864, 36452994894649858339584

Offset: 1

Ilya Gutkovskiy, Oct 06 2021

sign

Cf. A006973, A137852, A157159, A157164, A348206, A348207.

Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 - Sum_{n>=1} (-x)^n/n.

A348206 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + log(1 + x).

Original entry on oeis.org

1, -1, 5, -26, 204, -1434, 16862, -166536, 2377512, -29870400, 558727872, -8542202976, 185546362416, -3332732184768, 82441247589360, -1824937537167744, 50908186083448320, -1214743725939310848, 38998680958184088960, -1084067907183602910720, 36452994894649858339584

Offset: 1

Ilya Gutkovskiy, Oct 06 2021

sign

Cf. A006973, A137852, A157159, A157164, A348205, A348207.

Product_{n>=1} (1 + a(n)*x^n/n!) = 1 - Sum_{n>=1} (-x)^n/n.

cp's OEIS Frontend

A006973 Dimensions of representations by Witt vectors.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A147542 Product(1 + a(n)*x^n, n=1..infinity) = sum(F(k+1)*x^k, k=1..infinity) = 1/(1-x-x^2), where F(n) = A000045(n) (Fibonacci numbers).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A157159 Infinite product representation of series 1 - log(1-x) = 1 + Sum_{j>=1} (j-1)!*(x^j)/j!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A353607 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + sin(x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A353608 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + sinh(x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A353611 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + tan(x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A157164 1/product(1 - a(n)*(x^n)/n!, n=1..infinity) = 1 + sum((x^k)/k, k=1..infinity) = 1 - log(1-x).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A353609 Product_{n>=1} (1 + a(n)*x^(2*n)/(2*n)!) = cosh(x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A348205 Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + log(1 + x).

Views

Author

Keywords

Crossrefs

A147542 Product(1 + a(n)x^n, n=1..infinity) = sum(F(k+1)x^k, k=1..infinity) = 1/(1-x-x^2), where F(n) = A000045(n) (Fibonacci numbers).

A353609 Product_{n>=1} (1 + a(n)x^(2n)/(2*n)!) = cosh(x).