A141369 -id:A141369 - cp's OEIS frontend

A360987 E.g.f. A(x) satisfies A(x) = exp(x * A(-x)^2).

1, 1, -3, -23, 233, 3521, -62171, -1416407, 35880977, 1095318721, -36224195059, -1387587617239, 56675849155705, 2612993427672577, -127090039302776395, -6852033608852338199, 386750643197222855969, 23875394847093826450049

Offset: 0

Seiichi Manyama, Feb 27 2023

sign

Sum_{k=0..n} (2*n - 2*k + 1)^(k-1) * (2*k)^(n-k) * binomial(n,k) = (2*n+1)^(n-1) = A052750(n). - Vaclav Kotesovec, Jul 03 2025

Seiichi Manyama, Table of n, a(n) for n = 0..367

Cf. A052750, A141369, A196198, A360988, A360989, A360990.

Cf. A168600.

a(n) = sum(k=0, n, (2*n-2*k+1)^(k-1)*(-2*k)^(n-k)*binomial(n, k));

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

a(0) = 1; a(n) = (-1)^(n-1) * (n-1)! * Sum_{i, j, k>=0 and i+j+k=n-1} (-1)^i * (n-i) * a(i) * a(j) * a(k)/(i! * j! * k!). - Seiichi Manyama, Jul 06 2025

A360988 E.g.f. A(x) satisfies A(x) = exp(x * A(-x)^3).

Original entry on oeis.org

1, 1, -5, -44, 829, 14656, -488897, -13063616, 629051449, 22531502080, -1420908901469, -63859764079616, 4983153798630709, 269501734545522688, -25073583375908431769, -1585437525801020801024, 171326697778165116452977, 12401692280007001315999744

Offset: 0

Seiichi Manyama, Feb 27 2023

sign

Cf. A141369, A196198, A360987, A360989, A360990.

Cf. A168601.

a(n) = sum(k=0, n, (3*n-3*k+1)^(k-1)*(-3*k)^(n-k)*binomial(n, k));

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

a(0) = 1; a(n) = (-1)^(n-1) * (n-1)! * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} (-1)^i * (n-i) * a(i) * a(j) * a(k) * a(l)/(i! * j! * k! * l!). - Seiichi Manyama, Jul 06 2025

A385526 E.g.f. A(x) satisfies A(x) = exp(xA(3x)).

Original entry on oeis.org

1, 1, 7, 208, 23365, 9588976, 14040296659, 71747056999360, 1255862559932597257, 74168744207577385109248, 14599375893944236344767578111, 9483024632344097320792984610415616, 20158786175666520486280070249843236771213, 139271933359690469686747131442731382830399594496

Offset: 0

Seiichi Manyama, Jul 02 2025

nonn

Cf. A000272, A096538, A141369, A385527, A385528, A385529, A385530.

Cf. A015084.

nmax = 15; A[] = 1; Do[A[x] = E^(x*A[3*x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 02 2025 *)

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385526(n)
  A(3, n)
end

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

a(n) ~ c * n! * 3^(n*(n-1)/2), where c = 1.361839192264541770366149558100099215697354561... - Vaclav Kotesovec, Jul 02 2025

A196198 E.g.f. satisfies A(x) = exp(x/A(-x)).

Original entry on oeis.org

1, 1, 3, 4, -19, -64, 1207, 5440, -164071, -954368, 39943691, 284754944, -15250391099, -128749666304, 8402599565375, 81978198409216, -6309988001033167, -69853770233675776, 6194681665486634899, 76717804389440684032

Offset: 0

Paul D. Hanna, Sep 30 2011

sign

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 4*x^3/3! - 19*x^4/4! - 64*x^5/5! +...
where log(A(x)) = x/A(-x) begins:
x/A(-x) = x + 2*x^2/2! - 3*x^3/3! - 32*x^4/4! + 105*x^5/5! + 2016*x^6/6! - 10115*x^7/7! - 282624*x^8/8! +...+ n*A141369(n-1)*x^n/n! +...

Seiichi Manyama, Table of n, a(n) for n = 0..406

Cf. A141369.

Flatten[{1,1,3,Table[Sum[Binomial[n,k]*(n-k)^k*(-k+1)^(n-k-1),{k,0,n-1}],{n,3,20}]}] (* Vaclav Kotesovec, Feb 26 2014 *)

{a(n)=if(n==0,1,sum(k=0, n-1, binomial(n, k)*(n-k)^k*(-k+1)^(n-k-1)))}

{a(n)=local(A=1+x);for(i=1,n,A=exp(x/subst(A,x,-x+x*O(x^n))));n!*polcoeff(A,n)}

a(n) = Sum_{k=0..n-1} binomial(n,k) * (n-k)^k * (-k+1)^(n-k-1) for n>0 with a(0)=1.
E.g.f. satisfies:
_ A(x) = exp(x*exp(x/A(x))).
_ A(x) = exp(x* exp(x*exp(-x*exp(x*exp(-x*exp(x*exp(-x*...))))))).
_ A(x) = exp(x*B(x)) where B(x) = exp(x/B(x)) is the e.g.f. of A141369.
E.g.f. satisfies: x/exp(-x/A(x)) = log(A(x)). - Vaclav Kotesovec, Feb 26 2014
|a(n)| ~ c * n! / (n^(3/2) * r^n), where r = 0.5098636055230131449434409623392631606695606770070519241... is the root of the equation r*exp(1/LambertW(-I/r))/I = LambertW(-I/r), and c = 0.385745347287849929987791864025522098993432068... if n is even, and c = 0.12921599603996711137996765405025929272341118... if n is odd. - Vaclav Kotesovec, Feb 26 2014

A385527 E.g.f. A(x) satisfies A(x) = exp(xA(4x)).

Original entry on oeis.org

1, 1, 9, 457, 118961, 152894961, 940318147705, 26967408304580857, 3534888068831469959649, 2084993641133372935803249505, 5465706581663919414225671125834601, 63043356313898446097762231466174924913065, 3173076775252515207774429654590479617164788572049

Offset: 0

Seiichi Manyama, Jul 02 2025

nonn

Cf. A000272, A096538, A141369, A385526, A385528, A385529, A385530.

Cf. A015085.

nmax = 15; A[] = 1; Do[A[x] = E^(x*A[4*x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 02 2025 *)

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385527(n)
  A(4, n)
end

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * 4^k * binomial(n-1,k) * a(k) * a(n-1-k).

a(n) ~ c * n! * 2^(n*(n-1)), where c = 1.216702003338638031273833889488221691367428313263423339843... - Vaclav Kotesovec, Jul 02 2025

A385528 E.g.f. A(x) satisfies A(x) = exp(xA(-2x)).

Original entry on oeis.org

1, 1, -3, -47, 1385, 119601, -22345691, -10181013695, 10346973518097, 23934447308323873, -122307331801326167539, -1379021793666951568998159, 33874331587448813081748999673, 1804181313330860398948564389193681, -206892703326367302570264123699846971211

Offset: 0

Seiichi Manyama, Jul 02 2025

sign

Cf. A000272, A096538, A141369, A385526, A385527, A385529, A385530.

Cf. A015097.

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385528(n)
  A(-2, n)
end

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (-2)^k * binomial(n-1,k) * a(k) * a(n-1-k).

A385529 E.g.f. A(x) satisfies A(x) = exp(xA(-3x)).

Original entry on oeis.org

1, 1, -5, -152, 15949, 6548656, -9510189137, -48598095401792, 849885323784381337, 50192539805114962349824, -9878895951508580401879879229, -6416836884643090722807370469927936, 13640603845766595275775514993987722683941, 94239467260528503337471761892783659993298198528

Offset: 0

Seiichi Manyama, Jul 02 2025

sign

Cf. A000272, A096538, A141369, A385526, A385527, A385528, A385530.

Cf. A015098.

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385529(n)
  A(-3, n)
end

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

A385530 E.g.f. A(x) satisfies A(x) = exp(xA(-4x)).

Original entry on oeis.org

1, 1, -7, -359, 90705, 116586321, -715618113143, -20523234900205911, 2689857437569063003169, 1586566688643256394542888225, -4159073515698730238218108546470759, -47972197129364591236078587520718376138951, 2414519164037893898620724924577882591112859773297

Offset: 0

Seiichi Manyama, Jul 02 2025

sign

Cf. A000272, A096538, A141369, A385526, A385527, A385528, A385529.

Cf. A015099.

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385530(n)
  A(-4, n)
end

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (-4)^k * binomial(n-1,k) * a(k) * a(n-1-k).

A360989 E.g.f. satisfies A(x) = exp(x / A(-x)^2).

Original entry on oeis.org

1, 1, 5, 1, -231, 81, 55453, -40431, -30313231, 33477985, 29630916981, -43713004191, -45378051616631, 83666428734513, 100216964952070541, -221570666935625999, -301515678925659598623, 777062158771833364929, 1185517627245415533666277

Offset: 0

Seiichi Manyama, Feb 27 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..366

Cf. A141369, A196198, A360987, A360988, A360990.

a(n) = sum(k=0, n, (-2*n+2*k+1)^(k-1)*(2*k)^(n-k)*binomial(n, k));

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

A360990 E.g.f. satisfies A(x) = exp(x / A(-x)^3).

Original entry on oeis.org

1, 1, 7, -8, -827, 2896, 452179, -2511872, -560237303, 4254259456, 1237434920191, -11907540107264, -4275828959720435, 49800209789734912, 21288959122755516235, -290981680034059649024, -144324916601232035246831, 2264121148389579474141184

Offset: 0

Seiichi Manyama, Feb 27 2023

sign

Cf. A141369, A196198, A360987, A360988, A360989.

A360990 := proc(n)
    add((-3*n+3*k+1)^(k-1)*(3*k)^(n-k)*binomial(n,k),k=0..n) ;
end proc:
seq(A360990(n),n=0..60) ; # R. J. Mathar, Mar 12 2023

a(n) = sum(k=0, n, (-3*n+3*k+1)^(k-1)*(3*k)^(n-k)*binomial(n, k));

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

cp's OEIS Frontend

A360987 E.g.f. A(x) satisfies A(x) = exp(x * A(-x)^2).

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A360988 E.g.f. A(x) satisfies A(x) = exp(x * A(-x)^3).

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A385526 E.g.f. A(x) satisfies A(x) = exp(x*A(3*x)).

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A196198 E.g.f. satisfies A(x) = exp(x/A(-x)).

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A385527 E.g.f. A(x) satisfies A(x) = exp(x*A(4*x)).

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A385528 E.g.f. A(x) satisfies A(x) = exp(x*A(-2*x)).

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A385529 E.g.f. A(x) satisfies A(x) = exp(x*A(-3*x)).

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A385530 E.g.f. A(x) satisfies A(x) = exp(x*A(-4*x)).

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Ruby

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A360989 E.g.f. satisfies A(x) = exp(x / A(-x)^2).

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A385526 E.g.f. A(x) satisfies A(x) = exp(xA(3x)).

A385527 E.g.f. A(x) satisfies A(x) = exp(xA(4x)).

A385528 E.g.f. A(x) satisfies A(x) = exp(xA(-2x)).

A385529 E.g.f. A(x) satisfies A(x) = exp(xA(-3x)).

A385530 E.g.f. A(x) satisfies A(x) = exp(xA(-4x)).