Emanuele Munarini - cp's OEIS frontend

A374585 A family of marked Motzkin-like paths.

1, 1, 1, 1, 3, 9, 21, 41, 79, 169, 393, 913, 2051, 4537, 10165, 23257, 53759, 124153, 286009, 660161, 1531875, 3571753, 8348981, 19539209, 45792719, 107546633, 253153609, 597034609, 1410131683, 3334984025, 7897992213, 18730123449, 44476842431, 105740699609, 251664629689

Offset: 0

Emanuele Munarini, Jul 12 2024

Number of lattice paths from (0,0) to (n,0) that stay weakly in the first quadrant and such that each step is either H=(1,0), U=(2,1) or D=(2,-1), where the steps H and D can be marked H*, D*, so that (in the canonical decomposition) a marked step H* cannot be followed by the empty path or by H. For instance, a(5)=9 because we have HHHHH, HUD, HUD*, H*UD, H*UD*, UDH, UD*H, UHD and UHD*.

Cf. A023426.

CoefficientList[Series[((1-t)^2-Sqrt[1-4t+6t^2-4t^3-7t^4+8t^5])/(4t^4),{t,0,100}],t]
Table[Sum[Binomial[n-k,3k] 2^k CatalanNumber[k], {k,0,n/4}], {n,0,100}]

G.f: A(t) = ((1-t)^2-sqrt(1-4*t+6*t^2-4*t^3-7*t^4+8*t^5))/(4*t^4).
G.f. A(x) satisfies: A(t) = 1+t*A(t)+t*(A(t)-1-t*A(t))+2*t^4*A(t)^2, or 2*t^4*A(t)^2-(1-t)^2*A(t)+1-t = 0.
a(n) = Sum_{k=0..floor(n/4)} binomial(n-k,3*k)*2^k*Catalan(k).
Recurrence: a(n+4) = 2*a(n+3) - a(n+2) + 2*Sum_{k=0..n} a(k)*a(n-k).
D-finite with recurrence: (n+9)*a(n+5) -2*(2*n+15)*a(n+4) +6*(n+6)*a(n+3) -2*(2*n+9)*a(n+2) -7*(n+3)*a(n+1) +4*(2*n+3)*a(n)=0.
a(n) ~ (1/2)*sqrt((4-X)/(2*Pi))*X^(-n-2)/n^(3/2),
where X = 0.40355658567370456... is the unique positive real root of 8*x^3-7*x^2+4*x-1.

A296660 Expansion of the e.g.f. exp(-2x)/(1-4x).

Original entry on oeis.org

1, 2, 20, 232, 3728, 74528, 1788736, 50084480, 1602703616, 57697329664, 2307893187584, 101547300251648, 4874270412083200, 253462061428318208, 14193875439985836032, 851632526399150129152, 54504481689545608331264, 3706304754889101366394880

Offset: 0

Emanuele Munarini, Dec 18 2017

nonn

Binomial self-convolution of sequence A296618.

Cf. A001907, A056545, A097820, A296618.

CoefficientList[Series[Exp[-2x]/(1-4x),{x,0,12}],x]Range[0,12]!
Table[Sum[Binomial[n, k] 4^k k! (-2)^(n-k), {k, 0, n}], {n, 0, 12}]

makelist(sum(binomial(n,k)*4^k*k!*(-2)^(n-k),k,0,n),n,0,12);

x='x+O('x^99); Vec(serlaplace(exp(-2*x)/(1-4*x))) \\ Altug Alkan, Dec 18 2017

E.g.f.: exp(-2*x)/(1-4*x).
a(n) = Sum_{k=0..n} binomial(n,k)*4^k*k!*(-2)^(n-k).
Sum_{k=0..n} binomial(n,k)*2^(n-k)*a(k) = 4^n n!.
a(n+1)-4*(n+1)*a(n) = (-2)^(n+1).
D-finite with recurrence a(n+2)-(4*n+6)*a(n+1)-8*(n+1)*a(n) = 0.
From Vaclav Kotesovec, Dec 18 2017: (Start)
a(n) = exp(-1/2) * 4^n * Gamma(n + 1, -1/2).
a(n) ~ n! * exp(-1/2) * 4^n. (End)

A296618 Expansion of the e.g.f. exp(-x)/sqrt(1-4*x).

Original entry on oeis.org

1, 1, 9, 89, 1265, 22929, 506809, 13220521, 397585761, 13543386785, 515418398441, 21673889807481, 998003450868049, 49942515803293489, 2698849517019693465, 156631203355106962889, 9716434375682706344129, 641592631434102757993281

Offset: 0

Emanuele Munarini, Dec 17 2017

nonn

Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A052143.

A296618 := n -> (-1)^n*(I/2)*KummerU(1/2, n+3/2, -1/4):
seq(simplify(A296618(n)), n=0..17); # Peter Luschny, Dec 18 2017

Table[Sum[Binomial[n,k]Binomial[2k,k]k! (-1)^(n-k),{k,0,n}],{n,0,18}]
CoefficientList[Series[Exp[-x]/Sqrt[1-4x], {x,0,18}], x] Range[0,18]!

makelist(sum(binomial(n,k)*binomial(2*k,k)*k!*(-1)^(n-k),k,0,n),n,0,12);

x='x+O('x^99); Vec(serlaplace(exp(-x)/sqrt(1-4*x))) \\ Altug Alkan, Dec 17 2017

a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*k!*(-1)^(n-k).
a(n) = (i/2)*(-1)^n*U(1/2,n+3/2,-1/4), where U denotes the Kummer U function.
D-finite with recurrence: a(n+2) - (4*n+5)*a(n+1) - 4*(n+1)*a(n) = 0.
Sum_{k=0..n} binomial(n,k)*a(k)*a(n-k) = Sum_{k=0..n} binomial(n,k)*(-1)^(n-k)*2^(n+k)*k!.
Conjectures:
1) a(n+1) == a(n) (mod n) for all n >= 1.
2) a(n+k) == (-1)^k*a(n) (mod k) for all n and k >= 1.
a(n) ~ 2^(2*n + 1/2) * n^n / exp(n + 1/4). - Vaclav Kotesovec, Dec 17 2017
From Peter Bala, Jun 19 2023: (Start)
a(n) == 1 (mod 4).
Conjecture 1) above follows immediately from the stated recurrence equation. In fact, a(n+1) == a(n) (mod 8*n) for n >= 1.
For a proof of Conjecture 2) see the Bala link (corollary to Theorem 1, p. 5). (End)

A230053 Recurrence a(n+2) = (n+2)a(n+1)a(n), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 6, 48, 1440, 414720, 4180377600, 13869489586176000, 521817332305350780518400000, 72373400562952038729626622187536384000000000, 415422642927888257689749131592471020852730170822782196121600000000000000

Offset: 0

Emanuele Munarini, Oct 08 2017

nonn

Numbers of decimal digits in a(n) for 0 <= n <= 20: 1, 1, 1, 1, 2, 4, 6, 10, 17, 27, 44, 72, 117, 190, 307, 498, 806, 1305, 2112, 3417, 5530. - Robert Israel, Oct 09 2017

Robert Israel, Table of n, a(n) for n = 0..16

Cf. A000045.

f:= proc(n) option remember; n*procname(n-1)*procname(n-2) end proc:
f(0):= 1: f(1):= 1:
map(f, [$0..12]); # Robert Israel, Oct 08 2017

RecurrenceTable[{a[n + 2] == (n + 2) a[n + 1] a[n], a[0] == a[1] == 1}, a, {n, 0, 12}] (* or *)
Table[Product[(n - k + 1)^Fibonacci[k], {k, 0, n - 1}], {n, 0, 12}]

a(n) = Product_{k=0..n-1} (n-k+1)^Fibonacci(k).

a(n) ~ c^(phi^n) / n, where c = exp(Sum_{k>=1} log(k+1) / (sqrt(5)*phi^k) ) = 2.32072822997682611701924627353608916645018... and phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 05 2021, updated Mar 16 2025

A284604 Quadratic recurrence: a(n+2) = a(n+1)^2 + a(n)^2 + 1, with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 3, 11, 131, 17283, 298719251, 89233191216703091, 7962562414716697755180182566955283, 63402400208259008611705446682872670539115181497111590988296570564371

Offset: 0

Emanuele Munarini, Mar 30 2017

nonn

Cf. A000283.

[n le 2 select 1 else Self(n-1)^2+Self(n-2)^2+1: n in [1..10]]; // Bruno Berselli, Mar 30 2017

RecurrenceTable[{a[n + 2] == a[n + 1]^2 + a[n]^2 + 1, a[0] == 1, a[1] == 1}, a, {n, 0, 12}]
nxt[{a_,b_}]:={b,a^2+b^2+1}; NestList[nxt,{1,1},10][[;;,1]] (* Harvey P. Dale, Feb 16 2025 *)

a(n) := if (n=0 or n=1) then 1 else a(n-1)^2 + a(n-2)^2 + 1; makelist(a(n), n, 0, 12);

a(n+3) = a(n+2)^2 + a(n+2) - a(n)^2, with a(0) = a(1) = 1, a(2) = 3.

a(n) ~ c^(2^n), where c = 1.356519333072951374233963037913978335267300244021120535099185060013... - Vaclav Kotesovec, Apr 15 2017

A276536 Binomial sums of the cubes of the central binomial coefficients.

Original entry on oeis.org

1, 9, 233, 8673, 376329, 17800209, 890215361, 46294813497, 2478150328777, 135642353562321, 7556884938829233, 427106589765940137, 24429206859151618209, 1411391470651692285609, 82245902444586364980057, 4828398428680134702936273

Offset: 0

Emanuele Munarini, Nov 16 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..554
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind
The Wolfram Functions Site, Complete Elliptic Integrals, 2016.

Cf. Sum_{k = 0..n} binomial(n, k)*binomial(2*k, k)^m: A026375 (m=1), A248586 (m=2), this sequence (m=3).

[&+[Binomial(n, k)*Binomial(2*k, k)^3: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Nov 30 2016

Table[Sum[Binomial[n, k]Binomial[2k, k]^3, {k, 0, n}], {n, 0, 100}]

makelist(sum(binomial(n,k)*binomial(2*k,k)^3,k,0,n),n,0,12);

a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(2*k, k)^3.
Recurrence: (n^3 + 12n^2 + 48n + 64) * a(n+4) - (68n^3 + 714n^2 + 2500n + 2919) * a(n+3) + (198n^3 + 1782n^2 + 5363n + 5397) * a(n+2) - 98 * (2n^3 + 15n^2 + 37n + 30) * a(n+1) + 65 * (n^3 + 6n^2 + 11n + 6) * a(n) = 0.
G.f.: (4/Pi^2) * K(1/2 - 1/2 * sqrt((1-65*t)/(1-t)))^2 / (1-t), where K(x) is complete elliptic integral of the first kind (defined as in MathWorld or in The Wolfram Functions Site).
a(n) ~ 65^(n+3/2) / (512 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Nov 16 2016
a(n) = 4F3(1/2,1/2,1/2,-n; 1,1,1; -64). - Ilya Gutkovskiy, Nov 25 2016

A276537 Alternating binomial sums of the cubes of the central binomial coefficients.

Original entry on oeis.org

1, 7, 201, 7375, 312265, 14365887, 697859169, 35226348087, 1829569294665, 97138289500735, 5248514415816721, 287657066913117447, 15953440327189548001, 893653778439275931175, 50488236061157830951545, 2873526763346873838886815

Offset: 0

Emanuele Munarini, Nov 16 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..558
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind
The Wolfram Functions Site, Complete Elliptic Integrals, 2016.

Cf. A276536.

Similar sums of m-powers of the central binomial coefficients: A002426 (m=1), A278934 (m=2), this sequence (m=3).

[&+[(-1)^(n-k)*Binomial(n,k)*Binomial(2*k,k)^3: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Dec 03 2016

Table[Sum[Binomial[n,k]Binomial[2k,k]^3(-1)^(n-k),{k,0,n}],{n,0,100}]

makelist(sum(binomial(n,k)*binomial(2*k,k)^3*(-1)^(n-k),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*binomial(2*k,k)^3.
Recurrence: (n^3+12*n^2+48*n+64)*a(n+4)-(60*n^3+630*n^2+2204*n+2569)*a(n+3)-(186*n^3+1674*n^2+5037*n+5067)*a(n+2)-94*(2*n^3+15*n^2+37*n+30)*a(n+1)-63*(n^3+6*n^2+11*n+6)*a(n)=0.
G.f.: (4/Pi^2)*K(1/2-1/2*sqrt((1-63*t)/(1+t)))^2/(1+t), where K(x) is the complete elliptic integral of the first kind (defined as in MathWorld or in The Wolfram Functions Site).
a(n) ~ 3^(2*n+3) * 7^(n+3/2) / (512 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Nov 16 2016
a(n) = (-1)^n*4F3(1/2,1/2,1/2,-n; 1,1,1; 64). - Ilya Gutkovskiy, Nov 25 2016

A275822 Alternating sums of the cubes of the central binomial coefficients.

Original entry on oeis.org

1, 7, 209, 7791, 335209, 15667799, 773221225, 39651016343, 2092095886657, 112840936041343, 6193764391911873, 344853399798469695, 19429178297906958721, 1105629520934309041279, 63455683531507986958721, 3668895994183490904049279

Offset: 0

Emanuele Munarini, Nov 15 2016

nonn

Robert Israel, Table of n, a(n) for n = 0..555
The Wolfram Functions Site, Complete Elliptic Integrals

Cf. A079727.

L:= [seq((-1)^k*binomial(2*k,k)^3,k=0..20)]:
B:= ListTools:-PartialSums(L):
seq((-1)^(k+1)*B[k],k=1..nops(B)); # Robert Israel, Nov 21 2016

Table[Sum[Binomial[2 k, k]^3 (-1)^(n - k), {k, 0, n}], {n, 0, 20}]
Table[Sum[(-1)^(n - k) (k + 1)^3 CatalanNumber[k]^3, {k, 0, n}], {n, 0, 20}] (* Jan Mangaldan, Jul 07 2020 *)

makelist(sum(binomial(2*k,k)^3*(-1)^(n-k),k,0,n),n,0,12);

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k,k)^3); \\ Michel Marcus, Jul 07 2020

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2*k,k)^3.
Recurrence: (n+2)^3*a(n+2)-(3*n+4)*(21*n^2+66*n+52)*a(n+1)-8*(2n+3)^3*a(n)=0.
G.f.: (4/Pi^2)*K((1-sqrt(1-64*t))/2)^2/(1+t), where K(x) is complete elliptic integral of the first kind (defined as in The Wolfram Functions Site).
a(n) ~ 2^(6*n+6) / (65*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Nov 16 2016

A277638 Binomial partial sums of sequence A007004.

Original entry on oeis.org

1, 4, 37, 520, 8803, 165292, 3320023, 69943804, 1526981575, 34271322316, 786371041603, 18372739163632, 435772652437381, 10468369504009060, 254238148448959729, 6233226769739934964, 154092763036678601551, 3837301178450916902428, 96181503100227675085675

Offset: 0

Emanuele Munarini, Oct 25 2016

nonn

Cf. A007004.

Table[Sum[Binomial[n, k] Multinomial[k, k, k]/(k+1), {k, 0, n}], {n, 0, 100}]

makelist(sum(binomial(n, k)*multinomial_coeff(k, k, k)/(k+1), k, 0, n), n, 0, 12);

a(n) = Sum_{k=0..n} binomial(n,k) * multinomial(k,k,k)/(k+1).
a(n) = hypergeometric(1/3,2/3,-n; 1,2; -27).
a(n) == 1 (mod 3) for all natural n.
E.g.f.: exp(t) * hypergeometric(1/3,2/3; 1,2; 27*t).
From Vaclav Kotesovec, Oct 26 2016: (Start)
Recurrence: n*(n+1)*a(n) = 2*(3*n-1)*(5*n-3)*a(n-1) - (n-1)*(57*n-56)*a(n-2) + 28*(n-2)*(n-1)*a(n-3).
a(n) ~ 2^(2*n+3) * 7^(n+2) / (3^(11/2) * Pi * n^2).
(End)
Diff. eq. satisfied by the ordinary g.f.: t*(1-t)^2*(1-28*t)*A''(t)+2*(1-t)*(1-2*t)*(1-28*t)*A'(t)-2*(4-29*t+28*t^2)*A(t)=0. - Emanuele Munarini, Oct 28 2016
G.f.: hypergeom([1/3, 2/3],[2],27*x/(1-x))/(1-x). - Mark van Hoeij, Nov 28 2024

A277639 Double binomial partial sums of A007004.

Original entry on oeis.org

1, 4, 43, 718, 14779, 344452, 8725093, 234594766, 6596287411, 192032529388, 5747827847545, 175986201591130, 5490952102178725, 174077883157001740, 5594651323154783515, 181946073109880839450, 5978730547304013537475, 198263347772478727193740, 6628299876919271425393105, 223211734849614639629628010, 7566093949269408444819804937

Offset: 0

Emanuele Munarini, Oct 25 2016

nonn

Cf. A007004.

Table[Sum[Binomial[n, k]^2 Multinomial[k,k,k]/(k+1), {k,0,n}], {n,0,100}]

makelist(sum(binomial(n,k)^2*multinomial_coeff(k,k,k)/(k+1),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} binomial(n,k)^2*multinomial(k,k,k)/(k+1).
a(n) = hypergeometric(1/3,2/3,-n,-n;1,1,2;27).
Double e.g.f.: BesselI(0,2*sqrt(t))*hypergeometric(1/3,2/3;1,1,2;27*t).
D-finite with recurrence: n^2*(n+1)^2*(1058*n^4 - 7061*n^3 + 16158*n^2 - 14048*n + 3284)*a(n) = 2*n*(30682*n^7 - 219052*n^6 + 555798*n^5 - 545060*n^4 + 16565*n^3 + 323730*n^2 - 206943*n + 39408)*a(n-1) - (834762*n^8 - 7954803*n^7 + 30596846*n^6 - 59518007*n^5 + 57023894*n^4 - 13636388*n^3 - 20674168*n^2 + 16952656*n - 3600432)*a(n-2) + 2*(n-2)^2*(744832*n^6 - 5313736*n^5 + 13458434*n^4 - 12947434*n^3 - 64535*n^2 + 6504872*n - 2110473)*a(n-3) - 676*(n-3)^2*(n-2)^2*(1058*n^4 - 2829*n^3 + 1323*n^2 + 1317*n - 609)*a(n-4). - Vaclav Kotesovec, Oct 30 2016
a(n) ~ sqrt(205/162 + 1939/(729*sqrt(3))) * (28+6*sqrt(3))^n / (Pi^(3/2)*n^(5/2)). - Vaclav Kotesovec, Oct 30 2016

cp's OEIS Frontend

User: Emanuele Munarini

A374585 A family of marked Motzkin-like paths.

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A296660 Expansion of the e.g.f. exp(-2*x)/(1-4*x).

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A296618 Expansion of the e.g.f. exp(-x)/sqrt(1-4*x).

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A230053 Recurrence a(n+2) = (n+2)*a(n+1)*a(n), with a(0) = a(1) = 1.

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A284604 Quadratic recurrence: a(n+2) = a(n+1)^2 + a(n)^2 + 1, with a(0) = a(1) = 1.

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Magma

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A276536 Binomial sums of the cubes of the central binomial coefficients.

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Magma

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A276537 Alternating binomial sums of the cubes of the central binomial coefficients.

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Magma

Mathematica

Maxima

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A275822 Alternating sums of the cubes of the central binomial coefficients.

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A296660 Expansion of the e.g.f. exp(-2x)/(1-4x).

A230053 Recurrence a(n+2) = (n+2)a(n+1)a(n), with a(0) = a(1) = 1.