A073708 -id:A073708 - cp's OEIS frontend

A073709 First differences of A073708.

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]

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

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).

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.

A321354 Expansion of Product_{k>=0} (1 + x^(3^k))^(3^(k+1)).

Original entry on oeis.org

1, 3, 3, 10, 27, 27, 45, 108, 108, 147, 333, 333, 480, 1107, 1107, 1467, 3294, 3294, 3801, 8109, 8109, 9300, 19791, 19791, 22644, 48141, 48141, 50806, 104277, 104277, 107011, 216756, 216756, 224937, 458055, 458055, 454437, 905256, 905256, 870777, 1707075, 1707075

Offset: 0

Seiichi Manyama, Nov 07 2018

nonn

Also the coefficient of x^(3*n) in the expansion of Product_{k>=0} (1 + x^(3^k))^(3^k).

			Product_{k>=0} (1 + x^(3^k))^(3^k) = 1 + x + 3*x^3 + 3*x^4 + 3*x^6 + 3*x^7 + 10*x^9 + 10*x^10 + 27*x^12 + 27*x^13 + 27*x^15 + 27*x^16 + 45*x^18 + 45*x^19 + ... .

Cf. A073708, A321344, A321355, A321357.

A321355 Expansion of Product_{k>=0} (1 + x^(4^k))^(4^(k+1)).

Original entry on oeis.org

1, 4, 6, 4, 17, 64, 96, 64, 136, 480, 720, 480, 680, 2240, 3360, 2240, 2444, 7536, 11304, 7536, 7276, 21568, 32352, 21568, 21080, 62752, 94128, 62752, 62968, 189120, 283680, 189120, 178646, 525464, 788196, 525464, 454614, 1292992, 1939488, 1292992, 1085688, 3049760

Offset: 0

Seiichi Manyama, Nov 07 2018

nonn

Also the coefficient of x^(4*n) in the expansion of Product_{k>=0} (1 + x^(4^k))^(4^k).

			Product_{k>=0} (1 + x^(4^k))^(4^k) = 1 + x + 4*x^4 + 4*x^5 + 6*x^8 + 6*x^9 + 4*x^12 + 4*x^13 + 17*x^16 + 17*x^17 + 64*x^20 + 64*x^21 + 96*x^24 + 96*x^25 + ... .

Cf. A073708, A321345, A321354, A321357.

A321357 Expansion of Product_{k>=0} (1 + x^(5^k))^(5^(k+1)).

Original entry on oeis.org

1, 5, 10, 10, 5, 26, 125, 250, 250, 125, 325, 1500, 3000, 3000, 1500, 2600, 11500, 23000, 23000, 11500, 14950, 63250, 126500, 126500, 63250, 65905, 266275, 532550, 532550, 266275, 233480, 901125, 1802250, 1802250, 901125, 698425, 2591000, 5182000, 5182000, 2591000

Offset: 0

Seiichi Manyama, Nov 07 2018

nonn

Also the coefficient of x^(5*n) in the expansion of Product_{k>=0} (1 + x^(5^k))^(5^k).

			Product_{k>=0} (1 + x^(5^k))^(5^k) = 1 + x + 5*x^5 + 5*x^6 + 10*x^10 + 10*x^11 + 10*x^15 + 10*x^16 + 5*x^20 + 5*x^21 + 26*x^25 + 26*x^26 + ... .

Cf. A073708, A321354, A321355.

A321336 Expansion of Product_{k>=0} (1 - x^(2^k))^(2^(k+1)).

Original entry on oeis.org

1, -2, -3, 8, -6, 4, 26, -56, -7, 70, -51, 32, 120, -272, -200, 672, -182, -308, 1026, -1744, -660, 3064, -916, -1232, 2466, -3700, -3990, 11680, -1416, -8848, 13752, -18656, -8503, 35662, -14331, -7000, 27122, -47244, -29870, 106984, -25895, -55194, 140173, -225152

Offset: 0

Seiichi Manyama, Nov 05 2018

sign

Cf. A073708, A321327, A321335.

Equals the self-convolution of A321327.
G.f.: A(x) satisfies A(x) = ((1 - x) * A(x^2))^2, with A(0) = 1.
a(n) = A321327(2*n) for n >= 0.

cp's OEIS Frontend

A073709 First differences of A073708.

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

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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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A321354 Expansion of Product_{k>=0} (1 + x^(3^k))^(3^(k+1)).

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A321355 Expansion of Product_{k>=0} (1 + x^(4^k))^(4^(k+1)).

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A321357 Expansion of Product_{k>=0} (1 + x^(5^k))^(5^(k+1)).

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A321336 Expansion of Product_{k>=0} (1 - x^(2^k))^(2^(k+1)).

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