A073709 -id:A073709 - cp's OEIS frontend

A073710 Convolution of A073709, which is also the first differences of the unique terms of A073709.

1, 2, 7, 12, 35, 58, 133, 208, 450, 692, 1358, 2024, 3822, 5620, 10018, 14416, 25025, 35634, 59591, 83548, 136955, 190362, 303725, 417088, 655128, 893168, 1374632, 1856096, 2820456, 3784816, 5658968, 7533120, 11144042, 14754964, 21542374

Offset: 0

Paul D. Hanna, Aug 05 2002

First differences consist of duplicated terms: {1, 1, 5, 5, 23, 23, 75, 75, 242, 242, 666, 666, 1798, 1798, ...}; the convolution of these first differences results in: {1, 2, 11, 20, 81, 142, 451, 760, 2143, 3526, 8965, ...}, which in turn has first differences that consist of duplicated terms: {1, 1, 9, 9, 61, 61, 309, 309, ...}.

			(1 +x +3x^2 +3x^3 +10x^4 +10x^5 +22x^6 +22x^7 +57x^8 +57x^9 +...)^2 = (1 +2x +7x^2 +12x^3 +35x^4 +58x^5 +133x^6 +208x^7 +450x^8 +...) and the first differences of the unique terms {1,3,10,22,57,...} is {1,2,7,12,35,...}.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A073707, A073708, A073709.

a073710 n = a073710_list !! n
a073710_list = conv a073709_list [] where
   conv (v:vs) ws = (sum $ zipWith (*) ws' $ reverse ws') : conv vs ws'
                    where ws' = v : ws
-- Reinhard Zumkeller, Jun 13 2013

max = 70; f[x_] := Sum[ a[k]*x^k, {k, 0, max}]; a[0] = a[1] = 1; coes = CoefficientList[ Series[ f[x^2]^2 - (1 - x)*f[x], {x, 0, max}], x]; A073709 = Table[a[k], {k, 0, max}] /. Solve[ Thread[coes == 0]] // First; A073709 // Union // Differences // Prepend[#, 1]&

Let f(x) = sum_{n=0..inf} A073709(n) x^n, then f(x) satisfies f(x)^2 = sum_{n=0..inf} a(n) x^n, as well as the functional equation f(x^2)^2 = (1 - x)*f(x).

A073708 Generating function A(x) satisfies A(x) = (1+x)^2*A(x^2)^2, with A(0)=1.

Original entry on oeis.org

1, 2, 5, 8, 18, 28, 50, 72, 129, 186, 301, 416, 664, 912, 1368, 1824, 2730, 3636, 5234, 6832, 9788, 12744, 17724, 22704, 31506, 40308, 54730, 69152, 93592, 118032, 156888, 195744, 259625, 323506, 423021, 522536, 681642, 840748, 1083402, 1326056, 1705665

Offset: 0

Paul D. Hanna, Aug 04 2002

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 8*x^3 + 18*x^4 + 28*x^5 + 50*x^6 +...
where A(x)^2 = 1 + 4*x + 14*x^2 + 36*x^3 + 93*x^4 + 208*x^5 + 456*x^6 +...
This sequence equals the self-convolution of A073707, which begins:
[1, 1, 2, 2, 5, 5, 8, 8, 18, 18, 28, 28, 50, 50, ...].
The first differences of this sequence result in A073709:
[1, 1, 3, 3, 10, 10, 22, 22, 57, 57, 115, 115, ...];
the self-convolution of A073709 yields A073710:
[1, 2, 7, 12, 35, 58, 133, 208, ...],
which in turn equals the first differences of the unique terms of A073709.

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A073709, A073710. A073707(2n)=a(n).

a073708 n = a073708_list !! n
a073708_list = conv a073707_list [] where
   conv (v:vs) ws = (sum $ zipWith (*) ws' $ reverse ws') : conv vs ws'
                    where ws' = v : ws
-- Reinhard Zumkeller, Jun 13 2013

A073708list[n_] := Module[{m = 1, A = 1}, While[m <= n, m = 2 m; A = ((1 + x)*(A /. x -> x^2))^2] + O[x]^m; CoefficientList[A, x][[1 ;; n]]]; A073708list[50] (* Jean-François Alcover, Apr 21 2016, adapted from PARI *)

a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=((1+x)*subst(A,x,x^2))^2); polcoeff(A,n))

Equals the self-convolution of A073707.

Edited by Michael Somos, May 03 2003

Edited by Paul D. Hanna, Jan 04 2013

A073707 Coefficients of a power series whose convolution consists of only the even-indexed terms of the sequence.

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 8, 8, 18, 18, 28, 28, 50, 50, 72, 72, 129, 129, 186, 186, 301, 301, 416, 416, 664, 664, 912, 912, 1368, 1368, 1824, 1824, 2730, 2730, 3636, 3636, 5234, 5234, 6832, 6832, 9788, 9788, 12744, 12744, 17724, 17724, 22704, 22704, 31506, 31506

Offset: 0

Paul D. Hanna, Aug 04 2002

			(1 + x + 2x^2 + 2x^3 + 5x^4 + 5x^5 + 8x^6 + 8x^7 + 28x^8 + 28x^9 + ...)^2 = (1 + 2x + 5x^2 + 8x^3 + 18x^4 + 28x^5 + 50x^6 + 72x^7 + 129x^8 + ...).

Reinhard Zumkeller (confirmed by Paul D. Hanna), Table of n, a(n) for n = 0..10000
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See p. 18.

Cf. A073708, A073709.

a073707 n = a073707_list !! n
a073707_list = 1 : f 0 0 [1] where
   f x y zs = z : f (x + y) (1 - y) (z:zs) where
     z = sum $ zipWith (*) hzs (reverse hzs) where hzs = drop x zs
-- Reinhard Zumkeller, Dec 21 2011

nmax = 49; CoefficientList[ Series[ Product[ (1+x^(2^n))^(2^n), {n, 0, Log[nmax]/Log[2]}], {x, 0, nmax}], x] (* Jean-François Alcover, Jan 04 2013, from 2nd formula, modified by Vaclav Kotesovec, Oct 23 2020 *)

a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=(1+x)*subst(A,x,x^2)^2); polcoeff(A,n))

{a(n)=polcoeff(prod(k=0,#binary(n),(1+x^(2^k)+x*O(x^n))^(2^k)),n)}

G.f.: A(x) satisfies A(x) = (1+x)*A(x^2)^2, with A(0)=1.
G.f.: A(x) = Product_{n>=0} (1 + x^(2^n))^(2^n).
G.f.: A(x) = (1/(1 - x)) * Product_{n>=0} 1/(1 - x^(2^(n+1)))^(2^n). - Eitan Y. Levine, Jun 24 2023

Definition corrected by Paul D. Hanna, Feb 25 2010

Data corrected for n > 45 by Reinhard Zumkeller, Dec 21 2011

A321088 G.f. satisfies: A(x) = 1/(1 - x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 4, -1, 9, 3, 11, -4, 17, -2, 11, -24, 31, -3, 39, -35, 70, -14, 47, -107, 112, -27, 122, -163, 198, -90, 93, -409, 282, -108, 329, -487, 601, -160, 324, -1076, 835, -165, 907, -1298, 1478, -429, 565, -2973, 1745, -427, 1999, -3149, 3587, -528

Offset: 0

Seiichi Manyama, Nov 05 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Convolution inverse of A321326.

Cf. A067856, A073709, A129374.

b[n_] := If[n == 1, 1, Product[{p, e} = pe; If[2 == p, e--, If[e > 1, p = 0, p = -1]]; p^e, {pe, FactorInteger[n]}]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
a = etr[b];
a /@ Range[0, 100] (* Jean-François Alcover, Oct 01 2019 *)

Euler transform of A067856.
G.f.: Product_{k>0} 1/(1 - x^k)^A067856(k).
Product_{k>0} A(x^k) = Product_{k>=0} 1/(1 - x^(2^k))^(2^k). (Cf. A073709.)

A321335 Expansion of 1/(1 - x) * Product_{k>=0} 1/(1 - x^(2^k))^(2^(k+1)).

Original entry on oeis.org

1, 3, 10, 22, 57, 115, 248, 456, 906, 1598, 2956, 4980, 8802, 14422, 24440, 38856, 63881, 99515, 159106, 242654, 379609, 569971, 873696, 1290784, 1945912, 2839080, 4213712, 6069808, 8890264, 12675080, 18334048, 25867168, 37011210, 51766174, 73308548, 101638332, 142626458

Offset: 0

Seiichi Manyama, Nov 05 2018

nonn

Cf. A073708, A073709, A321336.

Partial sums of A073710.

G.f.: A(x) satisfies A(x) = ((1 + x) * A(x^2))^2 / (1 - x), with A(0) = 1.

a(n) = A073709(2*n) = A073709(2*n+1) for n >= 0.

A321327 Expansion of Product_{k>=0} (1 - x^(2^k))^(2^k).

Original entry on oeis.org

1, -1, -2, 2, -3, 3, 8, -8, -6, 6, 4, -4, 26, -26, -56, 56, -7, 7, 70, -70, -51, 51, 32, -32, 120, -120, -272, 272, -200, 200, 672, -672, -182, 182, -308, 308, 1026, -1026, -1744, 1744, -660, 660, 3064, -3064, -916, 916, -1232, 1232, 2466, -2466, -3700, 3700, -3990, 3990

Offset: 0

Seiichi Manyama, Nov 05 2018

sign

Convolution inverse of A073709.

Cf. A073707, A321326.

G.f.: A(x) satisfies A(x) = (1 - x) * A(x^2)^2, with A(0) = 1.

A357366 Expansion of Product_{k>=0} 1 / (1 - x^(2^k) - x^(2^(k+1)))^(2^k).

Original entry on oeis.org

1, 1, 4, 5, 18, 23, 59, 82, 203, 285, 610, 895, 1838, 2733, 5217, 7950, 14763, 22713, 40526, 63239, 110652, 173891, 297529, 471420, 796706, 1268126, 2116508, 3384634, 5606444, 8991078, 14791302, 23782380, 38955441, 62737821, 102388280, 165126101, 268844542, 433970643

Offset: 0

Ilya Gutkovskiy, Sep 25 2022

nonn

Cf. A073709, A084782, A173285, A237651.

nmax = 37; CoefficientList[Series[Product[1/(1 - x^(2^k) - x^(2^(k + 1)))^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
nmax = 37; A[] = 1; Do[A[x] = A[x^2]^2/(1 - x - x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = A(x^2)^2 / (1 - x - x^2).

a(n) ~ c * phi^(n+1) / sqrt(5), where c = Product_{k>=1} 1/(1 - x^(2^k) - x^(2^(k+1)))^(2^k) = 11.1991985012843182084779984477952870732899201240395056... and phi = A001622 is the golden ratio. - Vaclav Kotesovec, Oct 08 2022

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A073710 Convolution of A073709, which is also the first differences of the unique terms of A073709.

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A073708 Generating function A(x) satisfies A(x) = (1+x)^2*A(x^2)^2, with A(0)=1.

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A073707 Coefficients of a power series whose convolution consists of only the even-indexed terms of the sequence.

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A321088 G.f. satisfies: A(x) = 1/(1 - x) * Product_{k>0} A(x^(2*k)) / Product_{k>1} A(x^(2*k-1)).

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A321335 Expansion of 1/(1 - x) * Product_{k>=0} 1/(1 - x^(2^k))^(2^(k+1)).

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A321327 Expansion of Product_{k>=0} (1 - x^(2^k))^(2^k).

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A357366 Expansion of Product_{k>=0} 1 / (1 - x^(2^k) - x^(2^(k+1)))^(2^k).

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A321088 G.f. satisfies: A(x) = 1/(1 - x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).