A251592 -id:A251592 - cp's OEIS frontend

A369264 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+x^2)^3 ).

1, 3, 18, 127, 993, 8268, 71888, 645087, 5929527, 55544315, 528319662, 5088941628, 49539243900, 486606281496, 4816930145376, 48005470976271, 481262635723491, 4850084768085465, 49107197378659262, 499298960719688343, 5095861705240094097

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A369262, A369263.

Cf. A365843, A369270.

A369264 := proc(n)
    add(binomial(3*n+3,k) * binomial(4*n-2*k+2,n-2*k),k=0..floor(n/2)) ;
    %/(n+1) ;
end proc;
seq(A369264(n),n=0..70) ; # R. J. Mathar, Jan 25 2024

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3/(1+x^2)^3)/x)

a(n, s=2, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+3,k) * binomial(4*n-2*k+2,n-2*k).
D-finite with recurrence +18*n*(3*n+2)*(2*n+3)*(3*n+1) *(2355222972296552964811*n -2353391098681877598217) *(n+1)*a(n) +3*n*(10232370941059360726949011*n^5 -6279411058144420889732231*n^4 +26515854213844281466097465*n^3 -21761373746876376187551525*n^2 -12108806260534489559295636*n +3394771165638813123794516)*a(n-1) +2*(-132629080888282243656059365*n^6 +156440924520330612537351287*n^5 -1546737637908414661531599805*n^4 +6858652031514251350543113065*n^3 -10688884261686986291502236950*n^2 +6884443241518652198616376568*n -1531720470240397832109679200)*a(n-2) +16*(-488032865226571716800174339*n^6 +5743512241166673419623793625*n^5 -28798925871340480498482300305*n^4 +76975939990931613139744649055*n^3 -114305622490237072905660442676*n^2 +89044784395613178550071941760*n -28430479725567026023998437760)*a(n-3) +384*(3*n-7) *(3*n-8)*(17416466042177225377415141*n^4 -183745766144088004186571330*n^3 +680994833213916542429809801*n^2 -1015881953145852406207817800*n +470197111913757817462248180)*a(n-4) +9216*(n-4)*(3*n-7)*(3*n-10) *(85246481204976073615097*n -71936955710157680798041)*(3*n-8) *(3*n-11)*a(n-5)=0. - R. J. Mathar, Jan 25 2024
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^3 * (1+x^2)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369265 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3) ).

Original entry on oeis.org

1, 2, 7, 31, 153, 806, 4439, 25250, 147193, 874732, 5279635, 32276245, 199439761, 1243633652, 7815804351, 49455190791, 314807497953, 2014530780524, 12952334769203, 83628832755779, 542022781854953, 3525150296312984, 22998642171764363, 150478455899387966

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A369267, A369269.

Cf. A071969, A370247.

A369265 := proc(n)
    add(binomial(n+1,k) * binomial(3*n-3*k+1,n-3*k),k=0..floor(n/3)) ;
    %/(n+1) ;
end proc;
seq(A369265(n),n=0..70) ; # R. J. Mathar, Jan 25 2024

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^3))/x)

a(n, s=3, t=1, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+1,k) * binomial(3*n-3*k+1,n-3*k).
D-finite with recurrence 16*(n+1)*(2*n+1)*a(n) +4*(-89*n^2+15*n+2)*a(n-1) +3*(345*n^2-603*n+274)*a(n-2) +18*(-41*n^2+45*n+94)*a(n-3) +54*(-4*n^2+57*n-137)*a(n-4) +486*(n-4)*(n-5)*a(n-5) -243*(n-4)*(n-5)*a(n-6)=0. - R. J. Mathar, Jan 25 2024
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^3) )^(n+1). - Seiichi Manyama, Feb 14 2024

A369269 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3)^3 ).

Original entry on oeis.org

1, 2, 7, 33, 173, 962, 5586, 33498, 205846, 1289386, 8202247, 52845855, 344129832, 2261377872, 14976646685, 99863119809, 669860309538, 4517037850220, 30603008068997, 208211448723097, 1421986458302466, 9745007758311114, 66993247112160800

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A369268, A369270.

Cf. A369265, A369267.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^3)^3)/x)

a(n, s=3, t=3, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+3,k) * binomial(3*n-3*k+1,n-3*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A139526 Triangle A061356 read right to left.

Original entry on oeis.org

1, 1, 2, 1, 6, 9, 1, 12, 48, 64, 1, 20, 150, 500, 625, 1, 30, 360, 2160, 6480, 7776, 1, 42, 735, 6860, 36015, 100842, 117649, 1, 56, 1344, 17920, 143360, 688128, 1835008, 2097152, 1, 72, 2268, 40824, 459270, 3306744, 14880348, 38263752, 43046721, 1, 90, 3600, 84000, 1260000, 12600000, 84000000, 360000000, 900000000, 1000000000

Offset: 2

Alford Arnold, Apr 24 2008

Related to the two Appell sequences the Bernoulli polynomials B(n,x) and their umbral compositional inverses (cf. A074909) Up(n,x) = [(x+1)^(n+1)-x^(n+1)] / (n+1). With offset 0, the row polynomials of this entry P(n,x) = (Up(n,0))^(-n) * [x + Up(n,0)]^n = (n+1)^n * [x + 1/(n+1)]^n. Compare to the Abel polynomials of A061356, which are also an Appell sequence. - Tom Copeland, Nov 14 2014

			(1) times (1) = (1)
(1 1) * (1 2) = (1 2)
(1 2 1 ) * (1 3 9) = (1 6 9)
(1 3 3 1) * (1 4 16 64) = (1 12 48 64)
etc.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA. Second ed. 1994.
Peter D. Schumer (2004), Mathematical Journeys, page 168, Proposition 16.1 (c)

P. Bala, Fractional iteration of a series inversion operator
Wikipedia, Lambert W function

Cf. A000272 (row sums), A061356 (row reverse), A028421, A074909, A000169 (main diagonal), A251592, A260687.

A061356 := proc(n,k) binomial(n-2,k-1)*(n-1)^(n-k-1); end: A139526 := proc(n,k) A061356(n,n-k-1) ; end: for n from 2 to 14 do for k from 0 to n-2 do printf("%d,",A139526(n,k)) ; od: od: # R. J. Mathar, May 22 2008

T[n_, k_] := (n - 1)^k*Binomial[n - 2, n - k - 2];
Table[T[n, k], {n, 2, 11}, {k, 0, n - 2}] // Flatten (* Jean-François Alcover, Jun 13 2023 *)

for(n=2,12,forstep(k=n-1,1,-1,print1(binomial(n-2, k-1)*(n-1)^(n-k-1)","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 10 2008

E.g.f. (with offset 1) Sum_{n >= 1} (1 + n*t)^(n-1)*x^n/n! = x + (1 + 2*t)*x^2/2! + (1 + 6*t + 9*t^2)*x^3/3! + .... For properties of this function see Graham et al., equations 5.60, 5.61 and 7.71. The e.g.f. is the series reversion with respect to x of the function log(1 + x)/(1 + x)^t, which is the e.g.f. for a signed version of A028421. - Peter Bala, Jul 18 2013
From Peter Bala, Nov 16 2015: (Start)
E.g.f. with offset 0 and constant term 1: A(x,t) = ( Sum_{n >= 0} (n + 1)^(n-1)*t^n*x^n/n! )^(1/t). This is the generalized exponential series E_t(x) in the terminology of Graham et al., Section 5.4.
A(x,t)^m = 1 + Sum_{n >= 1} m*(m + n*t)^(n-1)*x^n/n!.
log(A(x,t)) = Sum_{n >= 1} (n*t)^(n-1)*x^n/n! = 1/t*T(t*x), where T(z) is Euler's tree function. See A000169.
A(x,t) = ( 1/x* Revert( x*exp(-x*t)) )^(1/t), where Revert is the series reversion operator with respect to x.
In the notation of the Bala link the e.g.f. is I^t(e^x), where I^t is a fractional series inversion operator. Cf. A251592, which has o.g.f. I^t(1 + x), and A260687, which has o.g.f. I^t(1/(1 - x)). (End)

More terms from R. J. Mathar and Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

A369262 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^3 ).

Original entry on oeis.org

1, 1, 5, 17, 80, 363, 1792, 8969, 46319, 242994, 1296046, 6996163, 38175142, 210162728, 1166020560, 6512854409, 36593709385, 206686641555, 1172856064443, 6683348391034, 38228129813288, 219411037878578, 1263245957786120, 7293833100110787, 42224142505632305

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A052709, A369226.

Cf. A369263, A369264.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+x^2)^3)/x)

a(n, s=2, t=3, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+3,k) * binomial(2*n-2*k,n-2*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^2)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369263 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2)^3 ).

Original entry on oeis.org

1, 2, 10, 54, 329, 2126, 14356, 100030, 713956, 5193064, 38354066, 286860714, 2168308302, 16537766036, 127114940840, 983657456878, 7657060437148, 59917814944376, 471062428422152, 3718952705982232, 29471640802526185, 234356062245289566, 1869405604537134116

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A218045, A369208.
Cf. A369262, A369264.
Cf. A370245.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^2)^3)/x)

a(n, s=2, t=3, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+3,k) * binomial(3*n-2*k+1,n-2*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^2)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369267 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3)^2 ).

Original entry on oeis.org

1, 2, 7, 32, 163, 884, 5009, 29310, 175750, 1074264, 6668825, 41929970, 266464579, 1708829584, 11044663663, 71871779008, 470495357634, 3096311833496, 20472771422946, 135937759368388, 906056228361095, 6059922934991008, 40657629626645463

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A369265, A369269.

Cf. A369266, A370249.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^3)^2)/x)

a(n, s=3, t=2, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+2,k) * binomial(3*n-3*k+1,n-3*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^3)^2 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369268 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^3)^3 ).

Original entry on oeis.org

1, 1, 2, 8, 29, 105, 414, 1695, 7046, 29853, 128644, 561262, 2474142, 11006108, 49343508, 222715440, 1011217425, 4615519083, 21165513228, 97467424198, 450541090701, 2089777230606, 9723511785608, 45371996501895, 212271904284993, 995513843930049, 4679212044797252

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A071969, A369266.

Cf. A369269, A369270.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+x^3)^3)/x)

a(n, s=3, t=3, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+3,k) * binomial(2*n-3*k,n-3*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369270 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+x^3)^3 ).

Original entry on oeis.org

1, 3, 15, 94, 657, 4902, 38233, 307953, 2541831, 21386810, 182754162, 1581699162, 13836248406, 122139271098, 1086638457429, 9733419373534, 87707244737511, 794505072627735, 7231017033165776, 66089527981542462, 606340568510978940, 5582088822346925210

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A369268, A369269.
Cf. A365843, A369264.
Cf. A369232.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3/(1+x^3)^3)/x)

a(n, s=3, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+3,k) * binomial(4*n-3*k+2,n-3*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^3 * (1+x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369296 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x^3)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 24, 84, 315, 1225, 4859, 19646, 80739, 336050, 1413587, 6000777, 25674462, 110598855, 479286932, 2088036939, 9139604421, 40174594432, 177267942918, 784889441217, 3486198469890, 15529021825140, 69355660644738, 310509670642611, 1393296782758244

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A063030, A369299.

Cf. A370274.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x^3)^2)/x)

a(n, s=3, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+k+1,k) * binomial(2*n-3*k,n-3*k).

a(n) = (1/(n+1)) * [x^n] 1/( (1-x) * (1-x^3)^2 )^(n+1). - Seiichi Manyama, Feb 14 2024

cp's OEIS Frontend

A369264 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+x^2)^3 ).

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A369265 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3) ).

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A369269 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3)^3 ).

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A139526 Triangle A061356 read right to left.

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A369262 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^3 ).

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A369263 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2)^3 ).

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A369267 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3)^2 ).

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A369268 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^3)^3 ).

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