A073707 -id:A073707 - cp's OEIS frontend

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

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

A073709 First differences of A073708.

Original entry on oeis.org

1, 1, 3, 3, 10, 10, 22, 22, 57, 57, 115, 115, 248, 248, 456, 456, 906, 906, 1598, 1598, 2956, 2956, 4980, 4980, 8802, 8802, 14422, 14422, 24440, 24440, 38856, 38856, 63881, 63881, 99515, 99515, 159106, 159106, 242654, 242654, 379609, 379609

Offset: 0

Paul D. Hanna, Aug 05 2002

The convolution of this sequence results in A073710 and is equal to the first differences of the unique terms of this sequence.

			G.f.: A(x) = 1 + x + 3*x^2 + 3*x^3 + 10*x^4 + 10*x^5 + 22*x^6 + 22*x^7 +...
where A(x) =  A(x^2)^2/(1-x) and thus
A(x) = 1 / [(1-x)*(1-x^2)^2*(1-x^4)^4*(1-x^8)^8*(1-x^16)^16*...].
Compare A(x)*(1-x) to A(x)^2:
A(x)*(1-x) = 1 + 2*x^2 + 7*x^4 + 12*x^6 + 35*x^8 + 58*x^10 + 133*x^12 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 12*x^3 + 35*x^4 + 58*x^5 + 133*x^6 + 208*x^7 +...
Also note that
A(x)^2/(1-x) = 1 + 3*x + 10*x^2 + 22*x^3 + 57*x^4 + 115*x^5 + 248*x^6 + 456*x^7 +...

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

Cf. A073707, A073708, A073710.

a073709 n = a073709_list !! n
a073709_list = 1 : zipWith (-) (tail a073708_list) a073708_list
--- Reinhard Zumkeller, Jun 13 2013

terms = 42; For[m = 1; A = 1, m <= 2*terms, m = 2*m, A = ((1+x)*(Normal[A] /. x -> x^2))^2 + O[x]^m]; Join[{1}, Differences[CoefficientList[A, x] ]][[1 ;; terms]] (* Jean-François Alcover, Mar 06 2013, updated Apr 23 2016 *)

{a(n)=polcoeff(prod(j=0,#binary(n),1/(1-x^(2^j)+x*O(x^n))^(2^j)),n)} \\ Paul D. Hanna, May 01 2010

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

G.f.: Product_{n>=0} 1/(1-x^(2^n))^(2^n). [Paul D. Hanna, May 01 2010]

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

Original entry on oeis.org

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).

A321325 G.f. satisfies: A(x) = (1 + x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, -2, 2, 1, 2, -3, 1, -3, -1, -7, 8, 4, 9, -7, 7, -7, 0, -21, 15, 2, 18, -23, 8, -25, -1, -43, 46, 17, 58, -34, 40, -41, 9, -98, 79, 10, 100, -98, 40, -123, -2, -191, 176, 43, 237, -136, 144, -192, 30, -362, 277, 12, 373, -314, 131, -457, -9, -606

Offset: 0

Seiichi Manyama, Nov 05 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A067856, A073707, A129373, A321326.

G.f.: Product_{k>0} (1 + x^k)^A067856(k).

Product_{k>0} A(x^k) = Product_{k>=0} (1 + x^(2^k))^(2^k). (Cf. A073707.)

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.

cp's OEIS Frontend

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

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A073709 First differences of A073708.

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

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A321325 G.f. satisfies: A(x) = (1 + x) * Product_{k>0} A(x^(2*k)) / Product_{k>1} A(x^(2*k-1)).

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

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A321325 G.f. satisfies: A(x) = (1 + x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).