A006973 -id:A006973 - cp's OEIS frontend

A156792 Triangle read by rows, T(n,k) = (A156791(n-k+1) * (A006973 * 0^(n-k))).

1, 1, 1, 6, 1, 2, 7, 6, 2, 9, 78, 7, 12, 9, 24, 420, 78, 14, 54, 24, 130, 6872, 420, 156, 63, 144, 130, 720, 17253, 6872, 840, 702, 168, 780, 720, 8505, 326552, 17253, 13744, 3780, 1872, 910, 4320, 8505, 35840

Offset: 0

Gary W. Adamson, Feb 15 2009

This is a convolution triangle: A006973 convolved with A156791.

As a property of eigentriangles, sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle:
       1,
       1,     1;
       6,     1,     2;
       7,     6,     2,    9;
      78,     7,    12,    9,   24;
     420,    78,    14    54,   24, 130;
    6872,   420,   156,   63,  144, 130,  720;
   17253,  6872,   840,  702,  168, 780,  720, 8505;
  326552, 17253, 13744, 3780, 1872, 910, 4320, 8505, 35840;
  ...
Example: Row 4 = (7, 6, 2, 9) = termwise products of (7, 6, 1, 1) and (1, 1, 2, 9).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A006973, A156791.

A006973[n_]:= A006973[n]= If[n<4, Max[n-1, 0], (n-1)!*(1 + Sum[k*(-A006973[k]/k!)^(n/k), {k, Most[Divisors[n]]}])];
S[n_, x_]:= Sum[A006973[j]*x^j, {j, 0, n+2}];
A156791:= With[{p=100}, CoefficientList[Series[S[p,x]/(x + S[p,x]), {x,0,p}], x]]
A156792[n_, k_]:= A156791[[n-k+2]]*(Boole[k==0] + A006973[k+1]);
Table[A156792[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 11 2021 *)

Triangle read by rows, T(n,k) = M*Q =(A156791(n-k+1) * (A006973 * 0^(n-k)))
M = an infinite lower triangular matrix with A156791: (1, 1, 6, 7, 78, ...) in every column.
Q = an infinite lower triangular matrix with A006973 prefaced with a 1 as the main diagonal: (1, 1, 2, 9, 24, 130, 720, 8505, ...) and the rest zeros.
Sum_{k=0..n} T(n, k) = A006973(n+1).
T(n, 0) = T(n+1, 1) = A156791(n).
T(n, k) = A156791(n-k+2)*([k=0] + A006973(k+1)). - G. C. Greubel, Jun 11 2021

Typo in last line of triangle corrected by Olivier Gérard, Aug 11 2016

A156791 INVERTi transform of A006973.

Original entry on oeis.org

1, 1, 6, 7, 78, 420, 6872, 17253, 326552, 2680988, 33949242, 386091406, 5974089338, 67562271804, 1708769998136, 16983594666421, 329723068729854, 5611760457560028, 121573316570762036, 2099347805249971662, 50720938223339101844, 1008798617234428297708

Offset: 1

Gary W. Adamson, Feb 15 2009

nonn

			The sequence begins (1, 1, ...); then for all further a(n), write (n-1) terms of A006973 backwards. Take the dot of the latter and the first (n-1) terms of sequence. Subtract from the next term of A006973.
Example: a(4) = 7 = (24 - (9, 2, 1) dot (1, 1, 6)) = 24 - 17.

G. C. Greubel, Table of n, a(n) for n = 1..440

Cf. A006973, A156792.

A006973[n_]:= A006973[n]= If[n<4, Max[n-1, 0], (n-1)!*(1 + Sum[k*(-A006973[k]/k!)^(n/k), {k, Most[Divisors[n]]}])];
S[n_, x_]:= Sum[A006973[j]*x^j, {j, 0, n+2}];
Rest@With[{p = 100}, CoefficientList[Series[S[p,x]/(x + S[p,x]), {x,0,60}], x]] (* G. C. Greubel, Jun 10 2021 *)

INVERTi transform of (A006973 with offset 1) = (1, 2, 9, 24, 130, 720, ...).

Coefficients of ( S(n, x)/(x + S(n, x)) ), where S(n, x) = Sum_{j=0..n+2} A006973(j)*x^j. - G. C. Greubel, Jun 10 2021

Terms a(11) onward added by G. C. Greubel, Jun 10 2021

A137852 G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = exp(x).

Original entry on oeis.org

1, 1, -2, 9, -24, 130, -720, 8505, -35840, 412776, -3628800, 42030450, -479001600, 7019298000, -82614884352, 1886805545625, -20922789888000, 374426276224000, -6402373705728000, 134987215801622184, -2379913632645120000, 55685679780013920000

Offset: 1

Paul D. Hanna, Feb 14 2008

Equals signed A006973 (except for initial term), where A006973 lists the dimensions of representations by Witt vectors.

			exp(x) = (1+x)*(1+x^2/2!)*(1-2*x^3/3!)*(1+9*x^4/4!)*(1-24*x^5/5!)* (1+130*x^6/6!)*(1-720*x^7/7!)*(1+8505*x^8/8!)*(1-35840*x^9/9!)*(1+412776*x^10/10!)*(1-3628800*x^11/11!)*...*(1+a(n)*x^n/n!)*...
Another recurrence: n=6; m=1,2,3=maxm(6)=A003056(6); fp(6,2) from {(1,5),(2,4)}, fp(6,3)=(1,2,3); a(6)= 1 - ( 6*a(1)*a(5) + 15*a(2)*a(4) + 60*a(1)*a(2)*a(3)). Check: 1 - (6*1*(-24) + 15*1*9 +60*1*1*(-2)) = 130 = a(6). - _Wolfdieter Lang_, Feb 20 2009

Alois P. Heinz, Table of n, a(n) for n = 1..170
Gottfried Helms, A dream of a (number-) sequence, 2007-2009.

Cf. A006973.

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

max = 22; f[x_] := Product[1 + a[n] x^n/n!, {n, 1, max}]; coes = CoefficientList[ Series[f[x] - Exp[x], {x, 0, max}], x]; sol = Solve[ Thread[coes == 0]][[1]]; Table[a[n] /. sol, {n, 1, max}] (* Jean-François Alcover, Nov 28 2011 *)
a[1] = 1; a[n_] := a[n] = (n-1)!*((-1)^n + Sum[d*(-a[d]/d!)^(n/d), {d, Divisors[n] ~Complement~ {1, n}}]);
Array[a, 30] (* Jean-François Alcover, Jan 11 2018 *)

{a(n)=if(n<1,0,if(n==1,1,(n-1)!*((-1)^n + sumdiv(n,d, if(d1, d*(-a(d)/d!)^(n/d))))))}
for(n=1,30,print1(a(n),", "))

/* As coefficients in product g.f.: */
{a(n)=if(n<1,0,n!*polcoeff(exp(x +x*O(x^n))/prod(k=0,n-1,1+a(k)*x^k/k! +x*O(x^n)),n))}
for(n=1,30,print1(a(n),", "))

a(n) = (n-1)!*[(-1)^n + Sum_{d divides n, 11 with a(1)=1.

Another 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 array in A036038) for any fp(n,m) from FP(n,m): a(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(n,m). Inputs a(1)=1, a(2)=1. See also array A008289(n,m) for the cardinality of the set FP(n,m). - Wolfdieter Lang, Feb 20 2009

A006173 Witt vector *2!.

Original entry on oeis.org

2, 1, 4, 13, 44, 135, 472, 1492, 5324, 17405, 63944, 215096, 799416, 2752909, 10310384, 36443256, 137263244, 489166324, 1860249448, 6739795717, 25596173800, 93596253769, 357974884304, 1319325363658, 5056389932088

Offset: 1

Simon Plouffe

nonn

If c is the Witt transform of b then b(n) = Sum_{d|n} A074650(n/d, c(d)).

The Somos transform sends sequence {a(n)} to sequence with g.f. Product_{i=1..n} 1/(1-a(i)*x^i).

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

H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-108.
H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-107. (Annotated scanned copy)

Cf. A006177, A006973.

Inverse Somos transform of A000108 shifted left. - Wouter Meeussen, Aug 20 2002

Witt transform of A060165. - Sean A. Irvine, Jan 15 2017

Edited by Christian G. Bower, Aug 20 2002

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

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

cp's OEIS Frontend

A156792 Triangle read by rows, T(n,k) = (A156791(n-k+1) * (A006973 * 0^(n-k))).

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A156791 INVERTi transform of A006973.

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A137852 G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = exp(x).

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A006173 Witt vector *2!.

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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).

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

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Maple

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

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Mathematica

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

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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).