A186996 -id:A186996 - cp's OEIS frontend

A182454 G.f. satisfies A(x) = 1 + xA(x) + x^2A(x)^5.

1, 1, 2, 7, 27, 112, 492, 2243, 10513, 50353, 245353, 1212398, 6061225, 30601910, 155808915, 799096655, 4124491215, 21408066097, 111672838857, 585128521777, 3078178384457, 16252057372887, 86089680204939, 457400940705274, 2436895852070559, 13015917111573039

Offset: 0

Paul D. Hanna, Apr 29 2012

nonn

Compare to a g.f. C(x) of Catalan numbers: C(x) = 1 + x*C(x) + x^2*C(x)^3.

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 27*x^4 + 112*x^5 + 492*x^6 +..
Related expansions:
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 141*x^4 + 615*x^5 + 2792*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 241*x^4 + 1080*x^5 + 4998*x^6 +...
A(x)^5 = 1 + 5*x + 20*x^2 + 85*x^3 + 380*x^4 + 1751*x^5 + 8270*x^6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A019497.

Cf. A000045, A000108, A001006, A186996.

{a(n)=local(A=1+x);for(i=1,n,A=1+x*A+x^2*A^5+x*O(x^n));polcoeff(A,n)}

{a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x*A^4)/((1+x*A^3)*1+x*O(x^n))); polcoeff(A, n)}

{a(n)=polcoeff((1/x)*serreverse(x^2/serreverse((sqrt(1+4*x-4*x^3+x^2*O(x^n))-1)/2)),n)}

{a(n)=polcoeff(sqrt((1/x)*serreverse((1+2*x-2*x^3-sqrt(1+4*x-4*x^3+x^3*O(x^n)))/(2*x))),n)}
for(n=0, 40, print1(a(n), ", "))

G.f.: A(x) = sqrt( (1/x)*Series_Reversion( (1 + 2*x - 2*x^3 - sqrt(1 + 4*x - 4*x^3))/(2*x) ) ).
G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A019497 (number of ternary search trees on n keys).
G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*A(x)^4) / (1 + x*A(x)^3).
Recurrence: 64*(n-1)*n*(2*n - 1)*(2*n + 1)*(5*n - 17)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(5*n - 9)*(5*n - 7)*a(n) = 32*(n-1)*(2*n - 1)*(5*n - 17)*(5*n - 14)*(5*n - 12)*(2500*n^5 - 16000*n^4 + 37400*n^3 - 38660*n^2 + 16767*n - 2304)*a(n-1) + (5*n - 17)*(5*n - 8)*(353125*n^8 - 4590625*n^7 + 26079625*n^6 - 84463075*n^5 + 169363570*n^4 - 212446228*n^3 + 159705192*n^2 - 64147968*n + 10184832)*a(n-2) + 8*(5*n - 2)*(1000000*n^9 - 19100000*n^8 + 158791250*n^7 - 752940875*n^6 + 2239835525*n^5 - 4325771435*n^4 + 5410989493*n^3 - 4216402206*n^2 + 1852118136*n - 348425280)*a(n-3) - 8*(5*n - 7)*(5*n - 4)*(5*n - 2)*(20000*n^7 - 368000*n^6 + 2847450*n^5 - 11988080*n^4 + 29592479*n^3 - 42711795*n^2 + 33256206*n - 10724400)*a(n-4) + 8*(n-4)*(2*n - 5)*(4*n - 19)*(4*n - 13)*(5*n - 12)*(5*n - 9)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 2)*a(n-5). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((1 + 2*r*s^4) / (10*Pi)) / (2*s * n^(3/2) * r^(n + 1/2)), where r = 0.1762643878022406506907195466376048222228890731329... and s = 1.517477187449684643254531724911215527841313263152... are roots of the system of equations 1 + r*s + r^2*s^5 = s, r + 5*r^2*s^4 = 1. - Vaclav Kotesovec, Nov 18 2017
a(n) = Sum_{k=0..floor(n/2)} binomial(n+3*k,k) * binomial(n+2*k,n-2*k) / (4*k+1). - Seiichi Manyama, Jul 26 2023

A364472 G.f. satisfies A(x) = 1 + xA(x) + x^2A(x)^6.

Original entry on oeis.org

1, 1, 2, 8, 35, 163, 808, 4162, 22041, 119325, 657384, 3673394, 20769983, 118610807, 683131766, 3963486380, 23144000681, 135911263309, 802143851323, 4755506884495, 28306896506651, 169110331570307, 1013643450123455, 6094125091837335, 36739933169338731

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000045, A000108, A001006, A182454, A186996, A364476.

Cf. A002295.

a(n) = sum(k=0, n\2, binomial(n+4*k, k)*binomial(n+3*k, n-2*k)/(5*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(n+4*k,k) * binomial(n+3*k,n-2*k) / (5*k+1) = Sum_{k=0..floor(n/2)} binomial(n+4*k,6*k) * binomial(6*k,k) / (5*k+1).

A364476 G.f. satisfies A(x) = 1 + xA(x) + x^2A(x)^7.

Original entry on oeis.org

1, 1, 2, 9, 44, 226, 1241, 7093, 41666, 250260, 1529993, 9488398, 59545909, 377451385, 2413157855, 15542535697, 100753850132, 656856027658, 4303970039402, 28328599504756, 187214549485759, 1241775795647609, 8263989319451514, 55163575187733922

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000045, A000108, A001006, A182454, A186996, A364472.

Cf. A002296, A364477.

a(n) = sum(k=0, n\2, binomial(n+5*k, k)*binomial(n+4*k, n-2*k)/(6*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(n+5*k,k) * binomial(n+4*k,n-2*k) / (6*k+1) = Sum_{k=0..floor(n/2)} binomial(n+5*k,7*k) * binomial(7*k,k) / (6*k+1).

A365733 G.f. satisfies A(x) = 1 + xA(x)(1 + x^5*A(x)^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 6, 16, 36, 71, 127, 215, 367, 676, 1376, 2982, 6514, 13855, 28407, 56543, 111127, 219918, 444450, 919744, 1933732, 4082467, 8576027, 17861347, 36938427, 76207797, 157652981, 328119005, 687377565, 1446665765, 3050094661, 6427116181

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A127902, A186996, A215340, A365079.

Cf. A002293, A365735.

a(n) = sum(k=0, n\6, binomial(n-5*k, k)*binomial(n-2*k+1, n-5*k)/(n-2*k+1));

a(n) = Sum_{k=0..floor(n/6)} binomial(n-5*k,k) * binomial(n-2*k+1,n-5*k) / (n-2*k+1) = Sum_{k=0..floor(n/6)} binomial(n-2*k,4*k) * binomial(4*k,k) / (3*k+1).

A378291 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3n-2r+k,r) * binomial(r,n-r)/(3n-2r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 6, 0, 1, 4, 9, 16, 20, 0, 1, 5, 14, 31, 56, 72, 0, 1, 6, 20, 52, 114, 208, 273, 0, 1, 7, 27, 80, 201, 438, 806, 1073, 0, 1, 8, 35, 116, 325, 800, 1739, 3220, 4333, 0, 1, 9, 44, 161, 495, 1341, 3260, 7077, 13168, 17869, 0

Offset: 0

Seiichi Manyama, Nov 21 2024

			Square array begins:
  1,   1,   1,    1,    1,    1,    1, ...
  0,   1,   2,    3,    4,    5,    6, ...
  0,   2,   5,    9,   14,   20,   27, ...
  0,   6,  16,   31,   52,   80,  116, ...
  0,  20,  56,  114,  201,  325,  495, ...
  0,  72, 208,  438,  800, 1341, 2118, ...
  0, 273, 806, 1739, 3260, 5615, 9119, ...

Columns k=0..1 give A000007, A186996.
Cf. A009766, A026300, A378289, A378290, A378292.
Cf. A378237.

T(n, k, t=1, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + x * A_k(x)^(3/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A186996.
B(x)^k = B(x)^(k-1) + x * B(x)^k + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-2,k+3) for n > 1.

A364410 G.f. A(x) satisfies A(x) = 1 + x^2 * (A(x) / (1 - x))^4.

Original entry on oeis.org

1, 0, 1, 4, 14, 52, 201, 800, 3260, 13536, 57068, 243664, 1051512, 4579088, 20097526, 88810872, 394811696, 1764477304, 7923087616, 35728412152, 161731039076, 734646128920, 3347600839252, 15298276784648, 70097391229500, 321974115549256, 1482242974320685

Offset: 0

Seiichi Manyama, Oct 15 2023

nonn

Partial sums give A186996.

Cf. A213336, A366645, A366646.

a(n) = sum(k=0, n\2, binomial(n+2*k-1, n-2*k)*binomial(4*k, k)/(3*k+1));

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

A365079 G.f. satisfies A(x) = 1 + xA(x)(1 + x^4*A(x)^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 6, 16, 36, 71, 131, 247, 511, 1156, 2696, 6172, 13664, 29563, 63871, 140341, 315185, 717962, 1639822, 3728276, 8432696, 19047924, 43166420, 98378502, 225355290, 517683270, 1190034046, 2735049866, 6287002806, 14467864356, 33355524916

Offset: 0

Seiichi Manyama, Aug 29 2023

nonn

Cf. A127902, A186996, A215340.

a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(n-k+1, n-4*k)/(n-k+1));

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * binomial(n-k+1,n-4*k)/(n-k+1).

cp's OEIS Frontend

A182454 G.f. satisfies A(x) = 1 + x*A(x) + x^2*A(x)^5.

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A364472 G.f. satisfies A(x) = 1 + x*A(x) + x^2*A(x)^6.

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A364476 G.f. satisfies A(x) = 1 + x*A(x) + x^2*A(x)^7.

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A365733 G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^5*A(x)^3).

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A378291 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3*n-2*r+k,r) * binomial(r,n-r)/(3*n-2*r+k) for k > 0.

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A364410 G.f. A(x) satisfies A(x) = 1 + x^2 * (A(x) / (1 - x))^4.

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A365079 G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^4*A(x)^3).

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A182454 G.f. satisfies A(x) = 1 + xA(x) + x^2A(x)^5.

A364472 G.f. satisfies A(x) = 1 + xA(x) + x^2A(x)^6.

A364476 G.f. satisfies A(x) = 1 + xA(x) + x^2A(x)^7.

A365733 G.f. satisfies A(x) = 1 + xA(x)(1 + x^5*A(x)^3).

A378291 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3n-2r+k,r) * binomial(r,n-r)/(3n-2r+k) for k > 0.

A365079 G.f. satisfies A(x) = 1 + xA(x)(1 + x^4*A(x)^3).