A112705 -id:A112705 - cp's OEIS frontend

A112707 Triangle built from partial sums of Catalan numbers multiplied by powers of nonpositive numbers.

1, 1, 1, 1, 0, 1, 1, 2, -1, 1, 1, -3, 7, -2, 1, 1, 11, -33, 16, -3, 1, 1, -31, 191, -119, 29, -4, 1, 1, 101, -1153, 1015, -291, 46, -5, 1, 1, -328, 7295, -9191, 3293, -579, 67, -6, 1, 1, 1102, -47617, 87037, -39715, 8171, -1013, 92, -7, 1, 1, -3760, 318463, -851186, 500957, -123079, 17131, -1623, 121

Offset: 0

Wolfdieter Lang, Oct 31 2005

The column sequences (without leading zeros) begin with A000012 (powers of 1), A032357(n)*(-1)^n, A064306(n)*(-1)^n, A112710, A112711, A113264-A113269, for m=0.. 10.

W. Lang: First 10 rows.

Row sums give A112708. Unsigned row sums give A112709.

Cf. A112705 (similar triangle with powers of positive numbers).

a(n, m)=sum(C(k)*(-m)^k, k=0..n-m), with C(k):=A000108(k) (Catalan) if n>m>0; a(n, n)=1, a(n, 0)=1, n>=0; a(n, m)=0 if n

G.f. for column m>=0 (without leading zeros): c(-m*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.

A112696 Partial sum of Catalan numbers A000108 multiplied by powers of 2.

Original entry on oeis.org

1, 3, 11, 51, 275, 1619, 10067, 64979, 431059, 2920403, 20119507, 140513235, 992530387, 7078367187, 50896392147, 368577073107, 2685777334227, 19678579249107, 144888698621907, 1071443581980627, 7954422715502547

Offset: 0

Wolfdieter Lang, Oct 31 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Third column (m=2) of triangle A112705.

a:=n->sum((binomial(2*j,j))*2^j/(j+1),j=0..n): seq(a(n), n=0..20); # Zerinvary Lajos, Oct 26 2006

Table[Sum[Binomial[2*j,j]*2^j/(j+1),{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 19 2012 *)

def A112696():
    f, c, n = 1, 1, 1
    while True:
        yield f
        n += 1
        c = c * (8*n - 12) // n
        f += c
a = A112696()
print([next(a) for  in range(21)]) # _Peter Luschny, Nov 30 2016

a(n) = Sum_{k=0..n} C(k)*2^k, n >= 0, with C(n):=A000108(n).
G.f.: c(2*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
a(n) = Sum_{j=0..n} binomial(2*j,j)*2^j/(j+1). - Zerinvary Lajos, Oct 26 2006
Recurrence: (n+1)*a(n) = 3*(3*n-1)*a(n-1) - 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(3*n+3)/(7*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

A112697 Partial sum of Catalan numbers (A000108) multiplied by powers of 3.

Original entry on oeis.org

1, 4, 22, 157, 1291, 11497, 107725, 1045948, 10428178, 106126924, 1097913928, 11511677470, 122057782762, 1306480339462, 14098243951822, 153208673236237, 1675240428936307, 18417589741637077, 203464608460961377

Offset: 0

Wolfdieter Lang, Oct 31 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Fourth column (m=3) of triangle A112705.

Cf. A000108.

CoefficientList[Series[(1-Sqrt[1-12*x])/(6*x)/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

x='x+O('x^50); Vec((1-sqrt(1-12*x))/(6*x*(1-x))) \\ G. C. Greubel, Mar 17 2017

a(n) = Sum_{k=0..n} C(k)*3^k, n>=0, with C(n) = A000108(n).
G.f.: c(3*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers (A000108).
Recurrence: (n+1)*a(n) = (13*n-5)*a(n-1) - 6*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 12^(n+1)/(11*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

A112698 Partial sum of (Catalan numbers A000108 multiplied by powers of 4).

Original entry on oeis.org

1, 5, 37, 357, 3941, 46949, 587621, 7616357, 101332837, 1375876965, 18987759461, 265554114405, 3755416368997, 53610591434597, 771525112379237, 11181285666076517, 163041321978836837, 2390321854565988197

Offset: 0

Wolfdieter Lang, Oct 31 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Fifth column (m=4) of triangle A112705.

CoefficientList[Series[(1-Sqrt[1-16*x])/(8*x)/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)
With[{nn=20},Accumulate[4^Range[0,nn] CatalanNumber[Range[0,nn]]]] (* Harvey P. Dale, Mar 11 2023 *)

x='x+O('x^50); Vec((1-sqrt(1-16*x))/(8*x*(1-x))) \\ G. C. Greubel, Mar 17 2017

a(n) = Sum_{k=0,..,n} C(k)*4^k, n>=0, with C(n):=A000108(n).
G.f.: c(4*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
Recurrence: (n+1)*a(n) = (17*n-7)*a(n-1) - 8*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 16^(n+1)/(15*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

Definition clarified by Harvey P. Dale, Mar 11 2023

A112699 Partial sum of Catalan numbers A000108 multiplied by powers of 5.

Original entry on oeis.org

1, 6, 56, 681, 9431, 140681, 2203181, 35718806, 594312556, 10090406306, 174113843806, 3044524000056, 53828703687556, 960689055250056, 17284175383375056, 313147365080640681, 5708299647795484431

Offset: 0

Wolfdieter Lang, Oct 31 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Sixth column (m=5) of triangle A112705.

CoefficientList[Series[(1-Sqrt[1-20*x])/(10*x)/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

x='x+O('x^50); Vec((1-sqrt(1-20*x))/(10*x*(1-x))) \\ G. C. Greubel, Mar 17 2017

a(n) = Sum_{k=0,..,n} C(k)*5^k, n>=0, with C(n):=A000108(n).
G.f.: c(5*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
Recurrence: (n+1)*a(n) = 3*(7*n-3)*a(n-1) - 10*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 20^(n+1)/(19*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

A112700 Partial sum of Catalan numbers A000108 multiplied by powers of 6.

Original entry on oeis.org

1, 7, 79, 1159, 19303, 345895, 6504487, 126597031, 2528447911, 51526205863, 1067116097959, 22394503831975, 475191351108007, 10177980935594407, 219758235960500647, 4778128782752211367, 104526001924311998887

Offset: 0

Wolfdieter Lang, Oct 31 2005

Robert Israel, Table of n, a(n) for n = 0..727

Seventh column (m=6) of triangle A112705.

ListTools:-PartialSums([seq(binomial(2*n,n)/(n+1)*6^n,n=0..50)]); # Robert Israel, Jun 28 2018

Accumulate[Table[ (CatalanNumber@ n)*6^n, {n, 0, 16}]] (* James C. McMahon, Jun 11 2024 *)

a(n) = Sum_{k=0..n} C(k)*6^k, with C(n):=A000108(n).
G.f.: c(6*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
Conjecture: (n+1)*a(n) +(-25*n+11)*a(n-1) +12*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 08 2016, verified by Robert Israel, Jun 28 2018
0 = a(n)*(+576*a(n+1) -636*a(n+2) +60*a(n+3)) +a(n+1)*(-564*a(n+1) +613*a(n+2) -61*a(n+3)) +a(n+2)*(+11*a(n+2) +a(n+3)) for all n>=0. - Michael Somos, Jun 28 2018

A112701 Partial sum of Catalan numbers (A000108) multiplied by powers of 7.

Original entry on oeis.org

1, 8, 106, 1821, 35435, 741329, 16270997, 369570944, 8613236374, 204812473608, 4949266755812, 121188396669810, 3000342229924222, 74979188061284522, 1888846103011564082, 47915719069874907917, 1222954711282739097587

Offset: 0

Wolfdieter Lang, Oct 31 2005

Robert Israel, Table of n, a(n) for n = 0..693

Column m=7 of triangle A112705.
Partial sums of A156266.
Cf. A000108, A000420.

f:= gfun:-rectoproc({(n+1)*a(n) +(-29*n+13)*a(n-1) +14*(2*n-1)*a(n-2)=0,a(0)=1,a(1)=8},a(n),remember):
map(f, [$0..50]); # Robert Israel, Aug 04 2020

CatalanNumber[#]*7^#& /@ Range[0, 20] // Accumulate (* Jean-François Alcover, Aug 29 2022 *)

a(n) = Sum_{k=0..n} A000108(k)*7^k.
G.f.: c(7*x)/(1-x), where c(x) = (1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
Conjecture: (n+1)*a(n) +(-29*n+13)*a(n-1) +14*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 08 2016
Conjecture verified using the d.e. (28*x^3-29*x^2+x)*y' + (42*x^2-16*x+1)*y=1 satisfied by the g.f. - Robert Israel, Aug 04 2020
a(n) = 7^n*binomial(2*n, n)*(1 - hypergeom([1, n+1/2], [n+2], 28))/(n+1) + (1 - 3*sqrt(3)*i)/14, where i denotes the imaginary units. - Stefano Spezia, Mar 31 2025

A112703 Partial sum of Catalan numbers A000108 multiplied by powers of 9.

Original entry on oeis.org

1, 10, 172, 3817, 95671, 2575729, 72725941, 2124619642, 63681430672, 1947319848190, 60511350647386, 1905278320822060, 60654011063307832, 1949006134928921932, 63131614948174818772, 2059214227480322203177

Offset: 0

Wolfdieter Lang, Oct 31 2005

Harvey P. Dale, Table of n, a(n) for n = 0..645

Tenth column (m=9) of triangle A112705.

Accumulate[Table[CatalanNumber[n]9^n,{n,0,20}]] (* Harvey P. Dale, Jun 22 2016 *)

a(n)=sum(C(k)*9^k, k=0..n), n>=0, with C(n):=A000108(n).
G.f.: c(9*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
Conjecture: +(n+1)*a(n) +(-37*n+17)*a(n-1) +18*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 08 2016

A112704 Partial sum of Catalan numbers A000108 multiplied by powers of 10.

Original entry on oeis.org

1, 11, 211, 5211, 145211, 4345211, 136345211, 4426345211, 147426345211, 5009426345211, 172969426345211, 6051569426345211, 214063569426345211, 7643063569426345211, 275087063569426345211, 9969932063569426345211

Offset: 0

Wolfdieter Lang, Oct 31 2005

Eleventh column (m=10) of triangle A112705.

a(n)=sum(C(k)*10^k, k=0..n), n>=0, with C(n):=A000108(n).

G.f.: c(10*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.

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A112706 Row sums of triangle A112705.

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A112707 Triangle built from partial sums of Catalan numbers multiplied by powers of nonpositive numbers.

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A112696 Partial sum of Catalan numbers A000108 multiplied by powers of 2.

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A112697 Partial sum of Catalan numbers (A000108) multiplied by powers of 3.

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A112698 Partial sum of (Catalan numbers A000108 multiplied by powers of 4).

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A112699 Partial sum of Catalan numbers A000108 multiplied by powers of 5.

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A112700 Partial sum of Catalan numbers A000108 multiplied by powers of 6.

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A112701 Partial sum of Catalan numbers (A000108) multiplied by powers of 7.

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A112703 Partial sum of Catalan numbers A000108 multiplied by powers of 9.

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