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Search: a321602 -id:a321602
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A303056
G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - A(x))^n.
+10
18
1, 1, 1, 8, 89, 1326, 24247, 521764, 12867985, 357229785, 11017306489, 373675921093, 13825260663882, 554216064798423, 23934356706763264, 1108017262467214486, 54747529760516714323, 2876096694574711401525, 160092696678371426933342, 9413031424290635395882462, 583000844360279565483710624
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
for any fixed integer k; here, k = 1 with r = 1, p = -A(x), q = (1+x). - Paul D. Hanna, Jun 22 2019
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^n - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.1610886538654... and c = 0.11739505492506... - Vaclav Kotesovec, Sep 26 2020
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 89*x^4 + 1326*x^5 + 24247*x^6 + 521764*x^7 + 12867985*x^8 + 357229785*x^9 + 11017306489*x^10 + ...
such that
1 = 1 + ((1+x) - A(x)) + ((1+x)^2 - A(x))^2 + ((1+x)^3 - A(x))^3 + ((1+x)^4 - A(x))^4 + ((1+x)^5 - A(x))^5 + ((1+x)^6 - A(x))^6 + ((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1+x)/(1 + (1+x)*A(x))^2 + (1+x)^4/(1 + (1+x)^2*A(x))^3 + (1+x)^9/(1 + (1+x)^3*A(x))^4 + (1+x)^16/(1 + (1+x)^4*A(x))^5 + (1+x)^25/(1 + (1+x)^5*A(x))^6 + (1+x)^36/(1 + (1+x)^6*A(x))^7 + ...
RELATED SERIES.
log(A(x)) = x + x^2/2 + 22*x^3/3 + 325*x^4/4 + 6186*x^5/5 + 137380*x^6/6 + 3478651*x^7/7 + 98674253*x^8/8 + 3096911434*x^9/9 + ...
PARTICULAR VALUES.
Although the power series A(x) diverges at x = -1/2, it may be evaluated formally.
Let t = A(-1/2) = 0.545218973635949431234950245034944106957612798888179456724264...
then t satisfies
(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.
(2) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).
Also,
A(r) = 1/2 at r = -0.54683649902292991492196620520872286547799291909992048564578...
where
(1) 1 = Sum_{n>=0} ( (1+r)^n - 1/2 )^n.
(2) 1 = Sum_{n>=0} (1+r)^(-n) / ( 1/(1+r)^n + 1/2 )^(n+1).
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^m - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A304642, A304639, A303926.
Cf. A321602, A321603, A321604, A321605.
Cf. A326282, A326283, A326284.
Cf. A337755, A337756, A337757.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 19 2018
STATUS
approved
A321603
G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(3*n) - A(x))^n.
+10
6
1, 3, 12, 235, 7872, 351924, 19340668, 1250971416, 92720438955, 7733929764167, 716488771114410, 72981787493017014, 8107675760704948748, 975749719762368998037, 126491959992115408069503, 17576241581408197850363955, 2606439876885873198662077692, 410925212330248782377865281826, 68641203626673300062880912740755, 12110976733338358608040713750036252
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(3*n) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(3*n^2) / (1 + (1+x)^(3*n)*A(x))^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 3*x + 12*x^2 + 235*x^3 + 7872*x^4 + 351924*x^5 + 19340668*x^6 + 1250971416*x^7 + 92720438955*x^8 + 7733929764167*x^9 + ...
such that
1 = 1 + ((1+x)^3 - A(x)) + ((1+x)^6 - A(x))^2 + ((1+x)^9 - A(x))^3 + ((1+x)^12 - A(x))^4 + ((1+x)^15 - A(x))^5 + ((1+x)^18 - A(x))^6 + ((1+x)^21 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1+x)^3/(1 + (1+x)^3*A(x))^2 + (1+x)^12/(1 + (1+x)^6*A(x))^3 + (1+x)^27/(1 + (1+x)^9*A(x))^4 + (1+x)^48/(1 + (1+x)^12*A(x))^5 + (1+x)^75/(1 + (1+x)^15*A(x))^6 + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 3 + 15*x + 624*x^2 + 28731*x^3 + 1638798*x^4 + 109462350*x^5 + 8333782509*x^6 + 710574703107*x^7 + 67015908514587*x^8 + ...
the coefficients of which are all divisible by 3:
(1/3) * A'(x)/A(x) = 1 + 5*x + 208*x^2 + 9577*x^3 + 546266*x^4 + 36487450*x^5 + 2777927503*x^6 + 236858234369*x^7 + 22338636171529*x^8 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(3*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A303056, A321602, A321604, A321605.
Cf. A326263.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 14 2018
STATUS
approved
A321604
G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.
+10
6
1, 4, 22, 564, 25157, 1499576, 109904860, 9480509576, 937113401201, 104240673195936, 12878161899791760, 1749261564410844864, 259132759251207789056, 41585481940418457992816, 7188476201158569394613976, 1331880173688346226092103696, 263358773243148578509342224153, 55363099822436514905885084770968, 12330972024423209530808891225876436, 2900976547500300324930009436969260936
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(4*n^2) / (1 + (1+x)^(4*n)*A(x))^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 4*x + 22*x^2 + 564*x^3 + 25157*x^4 + 1499576*x^5 + 109904860*x^6 + 9480509576*x^7 + 937113401201*x^8 + 104240673195936*x^9 + ...
such that
1 = 1 + ((1+x)^4 - A(x)) + ((1+x)^8 - A(x))^2 + ((1+x)^12 - A(x))^3 + ((1+x)^16 - A(x))^4 + ((1+x)^20 - A(x))^5 + ((1+x)^24 - A(x))^6 + ((1+x)^28 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1+x)^4/(1 + (1+x)^4*A(x))^2 + (1+x)^16/(1 + (1+x)^8*A(x))^3 + (1+x)^36/(1 + (1+x)^12*A(x))^4 + (1+x)^64/(1 + (1+x)^16*A(x))^5 + (1+x)^100/(1 + (1+x)^20*A(x))^6 + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 4 + 28*x + 1492*x^2 + 91788*x^3 + 6981484*x^4 + 621939700*x^5 + 63151305340*x^6 + 7181135905380*x^7 + 903210250234696*x^8 + ...
the coefficients of which are all divisible by 4:
(1/4) * A'(x)/A(x) = 1 + 7*x + 373*x^2 + 22947*x^3 + 1745371*x^4 + 155484925*x^5 + 15787826335*x^6 + 1795283976345*x^7 + 225802562558674*x^8 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(4*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A303056, A321602, A321603, A321605.
Cf. A326264.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 14 2018
STATUS
approved
A321605
G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.
+10
6
1, 5, 35, 1110, 61830, 4607001, 422112085, 45521033720, 5625206604320, 782244114339935, 120812011501389376, 20514224767917807795, 3798925417133114909240, 762102329400356260363990, 164678708686403817727101920, 38140958485665617437764886383, 9427520984195812306085385378080, 2477372683628569966077893189614835, 689743886246438120027048924784220410
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(5*n^2) / (1 + (1+x)^(5*n)*A(x))^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 5*x + 35*x^2 + 1110*x^3 + 61830*x^4 + 4607001*x^5 + 422112085*x^6 + 45521033720*x^7 + 5625206604320*x^8 + 782244114339935*x^9 + ...
such that
1 = 1 + ((1+x)^5 - A(x)) + ((1+x)^10 - A(x))^2 + ((1+x)^15 - A(x))^3 + ((1+x)^20 - A(x))^4 + ((1+x)^25 - A(x))^5 + ((1+x)^30 - A(x))^6 + ((1+x)^35 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1+x)^5/(1 + (1+x)^5*A(x))^2 + (1+x)^20/(1 + (1+x)^10*A(x))^3 + (1+x)^45/(1 + (1+x)^15*A(x))^4 + (1+x)^80/(1 + (1+x)^20*A(x))^5 + (1+x)^125/(1 + (1+x)^25*A(x))^6 + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 5 + 45*x + 2930*x^2 + 225545*x^3 + 21445630*x^4 + 2388480630*x^5 + 303204843520*x^6 + 43104182972905*x^7 + 6777636393880895*x^8 + ...
the coefficients of which are all divisible by 5:
(1/5) * A'(x)/A(x) = 1 + 9*x + 586*x^2 + 45109*x^3 + 4289126*x^4 + 477696126*x^5 + 60640968704*x^6 + 8620836594581*x^7 + 1355527278776179*x^8 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(5*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A303056, A321602, A321603, A321604.
Cf. A326265.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 14 2018
STATUS
approved
A326262
G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^(2*n) - A(x) )^n.
+10
5
1, 2, 7, 80, 1742, 51842, 1902589, 82219592, 4071164749, 226803165574, 14029472009781, 953926536359084, 70723894649169937, 5679305945331227594, 491179287055641264989, 45527108214667404725616, 4503148842172835722939285, 473502491643614888369261116, 52748299277043902326373361722, 6206479798643382507763241117360, 769187266152748793100664986340382, 100156538984193022704291755068539370
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..21.
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/(1-x)^(2*n) - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^(2*n)*A(x) )^n / (1-x)^(2*n^2).
(3) 1 = Sum_{n>=0} (1-x)^(2*n) / ( (1-x)^(2*n) + A(x) )^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 7*x^2 + 80*x^3 + 1742*x^4 + 51842*x^5 + 1902589*x^6 + 82219592*x^7 + 4071164749*x^8 + 226803165574*x^9 + 14029472009781*x^10 + ...
such that
1 = 1 + (1/(1-x)^2 - A(x)) + (1/(1-x)^4 - A(x))^2 + (1/(1-x)^6 - A(x))^3 + (1/(1-x)^8 - A(x))^4 + (1/(1-x)^10 - A(x))^5 + (1/(1-x)^12 - A(x))^6 + (1/(1-x)^14 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1-x)^2/((1-x)^2 + A(x))^2 + (1-x)^4/((1-x)^4 + A(x))^3 + (1-x)^6/((1-x)^6 + A(x))^4 + (1-x)^8/((1-x)^8 + A(x))^5 + (1-x)^10/((1-x)^10 + A(x))^6 + (1-x)^12/((1-x)^12 + A(x))^7 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1-x)^(-2*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A304639, A326263, A326264, A326265.
Cf. A321602.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 20 2019
STATUS
approved
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