A082137 -id:A082137 - cp's OEIS frontend

A082143 First subdiagonal of number array A082137.

1, 3, 20, 140, 1008, 7392, 54912, 411840, 3111680, 23648768, 180590592, 1384527872, 10650214400, 82158796800, 635361361920, 4924050554880, 38233804308480, 297374033510400, 2316387208396800, 18067820225495040, 141101072237199360, 1103153837490831360

Offset: 0

Paul Barry, Apr 06 2003

			a(0)=(2^(-1)+(0^0)/2)C(1,0)=2*(1/2)=1 (use 0^0=1).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A069723, A082144, A082145.

Cf. A000079, A001700, A069720.

a082143 0 = 1
a082143 n = (a000079 $ n - 1) * (a001700 n)
-- Reinhard Zumkeller, Jan 15 2015

[(2^(n-1) + 0^n/2)*Binomial(2*n+1,n): n in [0..30]]; // G. C. Greubel, Feb 05 2018

Join[{1}, Table[2^(n-1)* Binomial[2*n+1,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)

for(n=0,30, print1((2^(n-1) + 0^n/2)*Binomial(2*n+1,n), ", ")) \\ G. C. Greubel, Feb 05 2018

a(n) = (2^(n-1) + 0^n/2)*C(2n+1, n).
Conjecture: (n+1)*a(n) +4*(-2*n-1)*a(n-1)=0. - R. J. Mathar, Oct 19 2014
From Reinhard Zumkeller, Jan 15 2015: (Start)
a(n) = A000079(n-1) * A001700(n), for n > 0.
a(n) = A069720(n+1)/2. (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 64*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)) + 1/7.
Sum_{n>=0} (-1)^n/a(n) = 32*log(2)/27 - 1/9. (End)

A082144 A subdiagonal of number array A082137.

Original entry on oeis.org

1, 4, 30, 224, 1680, 12672, 96096, 732160, 5601024, 42997760, 331082752, 2556051456, 19778969600, 153363087360, 1191302553600, 9268801044480, 72219408138240, 563445537177600, 4401135695953920, 34414895667609600, 269374774271016960, 2110381254330286080

Offset: 0

Paul Barry, Apr 06 2003

			a(0)=(2^(-1)+(0^0)/2)C(2,0)=2*(1/2)=1 (use 0^0=1).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A069723, A082137, A082143, A082145.

[(2^(n-1) + 0^n/2)*Binomial(2*n+2,n): n in [0..30]]; // G. C. Greubel, Feb 05 2018

Join[{1}, Table[2^(n-1)*Binomial[2*n+2, n], {n,1,50}]] (* G. C. Greubel, Feb 05 2018 *)

for(n=0,30, print1((2^(n-1) + 0^n/2)*Binomial(2*n+2,n), ", ")) \\ G. C. Greubel, Feb 05 2018

a(n) = (2^(n-1) + 0^n/2)*C(2*n+2, n).
(n+2)*a(n) +12*(-n-1)*a(n-1) +16*(2*n-1)*a(n-2)=0. - R. J. Mathar, Oct 29 2014
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 88*arccot(sqrt(7))/(7*sqrt(7)) - 3/7.
Sum_{n>=0} (-1)^n/a(n) = 52*log(2)/27 - 5/9. (End)

A082145 A subdiagonal of number array A082137.

Original entry on oeis.org

1, 5, 42, 336, 2640, 20592, 160160, 1244672, 9674496, 75246080, 585761792, 4564377600, 35602145280, 277970595840, 2172375244800, 16992801914880, 133035751833600, 1042374243778560, 8173537721057280, 64136851016908800, 503613708419727360, 3956964851869286400

Offset: 0

Paul Barry, Apr 06 2003

			a(0) = ( 2^(-1)+(0^0)/2 )*C(3,0) = ( 1/2+1/2 )*1 = 1 (use 0^0 = 1). - clarified by _Jon Perry_, Oct 29 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A069723, A082143, A082144.

[(2^(n-1)+(0^n)/2)*Binomial(2*n+3, n): n in [0..30]]; // Vincenzo Librandi, Oct 30 2014

Z:=(1-3*z-sqrt(1-4*z))/sqrt(1-4*z)/64: Zser:=series(Z, z=0, 32): seq(coeff(Zser*2^(n+1), z, n), n=4..23); # Zerinvary Lajos, Jan 01 2007

Join[{1}, Table[2^(n-1)* Binomial[2*n+3,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)

for(n=0,30, print1((2^(n-1) + 0^n/2)*Binomial(2*n+3,n), ", ")) \\ G. C. Greubel, Feb 05 2018

a(n) = ( 2^(n-1) + (0^n)/2 )*binomial(2*n+3, n).
(n+3)*a(n) +2*(-7*n-13)*a(n-1) +24*(2*n+1)*a(n-2)=0. - R. J. Mathar, Oct 29 2014
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 37/7 - 208*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 296*log(2)/27 - 61/9. (End)

A134308 A007318 * A082137.

Original entry on oeis.org

1, 2, 1, 4, 4, 2, 8, 12, 12, 4, 16, 32, 48, 32, 8, 32, 80, 160, 160, 80, 16, 64, 192, 480, 640, 480, 192, 32, 128, 448, 1344, 2240, 2240, 1344, 448, 64, 256, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 512, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256

Offset: 0

Gary W. Adamson, Oct 19 2007

row sums = A007582: (1, 3, 10, 36, 136, 528, ...).

			First few rows of the triangle:
   1;
   2,  1;
   4,  4,   2;
   8, 12,  12,   4;
  16, 32,  48,  32,  8;
  32, 80, 160, 160, 80, 16;
  ...

Cf. A007318, A007582, A082137.

Binomial transform of A082137, as infinite lower triangular matrices.

a(5), a(46) corrected, a(27), a(34) split by Georg Fischer, Jun 01 2023

A069720 a(n) = 2^(n-1)binomial(2n-1, n).

Original entry on oeis.org

1, 6, 40, 280, 2016, 14784, 109824, 823680, 6223360, 47297536, 361181184, 2769055744, 21300428800, 164317593600, 1270722723840, 9848101109760, 76467608616960, 594748067020800, 4632774416793600, 36135640450990080, 282202144474398720, 2206307674981662720, 17266755717247795200

Offset: 1

Valery A. Liskovets, Apr 07 2002

Number of rooted unicursal planar maps with n edges (unicursal means that exactly two nodes are of odd valency; there is an Eulerian path).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Harlan J. Brothers, Pascal's Prism: Supplementary Material.
Valery A. Liskovets and Timothy R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

First superdiagonal of number array A082137.

Cf. A000079, A001700, A003584, A069723, A069724, A082143.

a069720 n = (a000079 $ n - 1) * (a001700 $ n - 1)
-- Reinhard Zumkeller, Jan 15 2015

[2^(n-2)*Binomial(2*n, n): n in [1..25]]; // Vincenzo Librandi, Apr 14 2018

Z:=(1-sqrt(1-2*z))*4^(n-1)/sqrt(1-2*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..20); # Zerinvary Lajos, Jan 01 2007

Table[2^(n-1) Binomial[2n-1,n],{n,20}] (* Harvey P. Dale, Jan 20 2013 *)

a(n) = binomial(2*n-1,n)<<(n-1) \\ Charles R Greathouse IV, Feb 06 2017

def A069720(n): return 2^(n-2)*binomial(2*n, n)
print([A069720(n) for n in range(1,31)]) # G. C. Greubel, Jan 18 2025

a(n) = 2^(n-2)*binomial(2*n, n).
G.f.: (1-sqrt(1-8*x))/(4*x*sqrt(1-8*x)) = 2/(sqrt(1-8*x)*(1-sqrt(1-8*x))) - 1/(2*x). - Paul Barry, Sep 06 2004
D-finite with recurrence: n*a(n) - 4*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Apr 01 2012
E.g.f.: a(n) = n! * [x^n] (exp(4*x)*BesselI(0, 4*x) - 1)/4. - Peter Luschny, Aug 25 2012
From Reinhard Zumkeller, Jan 15 2015: (Start)
a(n) = A000079(n-1) * A001700(n-1); for n > 1:
a(n) = 2*A082143(n-1). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=1} 1/a(n) = 4/7 + 32*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 + 16*log(2)/27. (End)
a(n) = ((2*n)!/4) * [x^n] (BesselI(0, 2*sqrt(2*x)) - 1). - G. C. Greubel, Jan 18 2025

A069723 a(n) = 2^(n-1)binomial(2n-3, n-1).

Original entry on oeis.org

1, 2, 12, 80, 560, 4032, 29568, 219648, 1647360, 12446720, 94595072, 722362368, 5538111488, 42600857600, 328635187200, 2541445447680, 19696202219520, 152935217233920, 1189496134041600, 9265548833587200, 72271280901980160, 564404288948797440, 4412615349963325440

Offset: 1

Valery A. Liskovets, Apr 07 2002

Number of rooted unicursal planar maps with n edges and two vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Harlan J. Brothers, Pascal's Prism: Supplementary Material.
Valery A. Liskovets and Timothy R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Main diagonal of array A082137.

Cf. A069720, A069721, A069722, A082143, A082144, A082145, A088218.

Z:=(1-sqrt(1-z))*8^n/sqrt(1-z)/2: Zser:=series(Z, z=0, 33): seq(coeff(Zser, z, n), n=0..20); # Zerinvary Lajos, Jan 16 2007

Table[2^(n - 1) * Binomial[2*n - 3, n - 1], {n, 1,50}] (* G. C. Greubel, Jan 15 2017 *)

# Assuming offset 0:
A069723  = lambda n: (rising_factorial(n, n)/factorial(n)) << n
[A069723(n) for n in (0..20)] # Peter Luschny, Nov 30 2014

a(n) = A069722(n)/2, n>1.
G.f.: 4*x/(sqrt(1-8*x) * (1-sqrt(1-8*x))). - Paul Barry, Sep 06 2004
With offset 0: a(n) = (0^n + 2^n*binomial(2n, n))/2. - Paul Barry, Sep 24 2004
D-finite with recurrence (-n+1)*a(n) + 4*(2*n-3)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
With offset 0: a(n) = 2^n*rf(n,n)/n! = 2^n*A088218(n), where rf denotes the rising factorial. - Peter Luschny, Nov 30 2014
a(n) = Sum_{k=0..n} binomial(n+k-1,k)*binomial(2*n-1, n-k). - Vladimir Kruchinin, Nov 11 2016
a(n) ~ 2^(3*n-4)/sqrt(Pi*n). - Ilya Gutkovskiy, Nov 11 2016
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=1} 1/a(n) = 9/7 + 16*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7/9 - 8*log(2)/27. (End)

A134309 Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gary W. Adamson, Oct 19 2007

As infinite lower triangular matrices, binomial transform of A134309 = A082137. A134309 * A007318 = A055372. A134309 * [1,2,3,...] = A057711: (1, 2, 6, 16, 40, 96, 224,...).

Triangle read by rows given by [0,0,0,0,0,0,0,0,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2007

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 0, 4;
  0, 0, 0, 0, 8;
  0, 0, 0, 0, 0, 16;
  ...

Cf. A011782 (diagonal elements: 1 followed by 1, 2, 4, 8, ... = A000079: 2^n).

Cf. A055372, A057711, A082137.

Join[{1},Flatten[Table[Join[{PadRight[{},n],2^(n-1)}],{n,20}]]] (* Harvey P. Dale, Jan 04 2024 *)

A134309(r,c)=if(r==c,2^max(r-1,0),0) \\ M. F. Hasler, Mar 29 2022

Triangle, T(0,0) = 1, then for n > 0, n zeros followed by 2^(n-1). Infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main diagonal and the rest zeros.
G.f.: (1 - y*x)/(1 - 2*y*x). - Philippe Deléham, Feb 04 2012
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Feb 04 2012
Diagonal is A011782, other elements are 0. - M. F. Hasler, Mar 29 2022

A082139 A transform of binomial(n,5).

Original entry on oeis.org

1, 6, 42, 224, 1008, 4032, 14784, 50688, 164736, 512512, 1537536, 4472832, 12673024, 35094528, 95256576, 254017536, 666796032, 1725825024, 4410441728, 11142168576, 27855421440, 68975329280, 169303080960, 412216197120

Offset: 0

Paul Barry, Apr 06 2003

Sixth row of number array A082137. C(n,5) has e.g.f. (x^5/5!)exp(x). The transform averages the binomial and inverse binomial transforms.

			a(0) = (2^(-1) + 0^0/2)*binomial(5,0) = 2*(1/2) = 1 (use 0^0 = 1).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Cf. A080951, A080952, A082138.

Cf. A082140, A082141, A082138, A080951, A080929, A057711.

[(Ceiling(Binomial(n+5, 5)*2^(n-1))) : n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

[seq (ceil(binomial(n+5,5)*2^(n-1)),n=0..23)]; # Zerinvary Lajos, Nov 01 2006

Drop[With[{nmax = 56}, CoefficientList[Series[x^5*Exp[x]*Cosh[x]/5!, {x, 0, nmax}], x]*Range[0, nmax]!], 5] (* or *) Join[{1}, Table[2^(n-1)* Binomial[n+5,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)

my(x='x+O('x^30)); Vec(serlaplace(x^5*exp(x)*cosh(x)/5!)) \\ G. C. Greubel, Feb 05 2018

Equals 2 * A080952.
a(n) = (2^(n-1) + 0^n/2)*C(n+5, n).
a(n) = Sum_{j=0..n} C(n+5, j+5)*C(j+5, 5)*(1+(-1)^j)/2.
G.f.: (1 -6*x +30*x^2 -80*x^3 +120*x^4 -96*x^5 +32*x^6)/(1-2*x)^6.
E.g.f.: x^5*exp(x)*cosh(x)/5! (preceded by 5 zeros).
a(n) = ceiling(binomial(n+5,5)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 20*log(2) - 38/3.
Sum_{n>=0} (-1)^n/a(n) = 1620*log(3/2) - 656. (End)

A082140 A transform of binomial(n,6).

Original entry on oeis.org

1, 7, 56, 336, 1680, 7392, 29568, 109824, 384384, 1281280, 4100096, 12673024, 38019072, 111132672, 317521920, 889061376, 2444918784, 6615662592, 17641766912, 46425702400, 120706826240, 310388981760, 790081044480

Offset: 0

Paul Barry, Apr 06 2003

Seventh row of number array A082137. C(n,6) has e.g.f. (x^6/6!)exp(x). The transform averages the binomial and inverse binomial transforms.

			a(0) = (2^(-1) + 0^0/2)*binomial(6,0) = 2*(1/2) = 1 (use 0^0 = 1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Cf. A082137, A082139, A080951.

For n>0, a(n) = 1/2 * A002409(n).

[(2^(n-1) + 0^n/2)*Binomial(n+6,n): n in [0..30]]; // G. C. Greubel, Feb 05 2018

[seq (ceil(binomial(n+6,6)*2^(n-1)),n=0..22)]; # Zerinvary Lajos, Nov 01 2006

Drop[With[{nmax = 56}, CoefficientList[Series[x^6*Exp[x]*Cosh[x]/6!, {x, 0, nmax}], x]*Range[0, nmax]!], 5] (* or *) Join[{1}, Table[2^(n-1)* Binomial[n+6,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)
LinearRecurrence[{14,-84,280,-560,672,-448,128},{1,7,56,336,1680,7392,29568,109824},30] (* Harvey P. Dale, Jul 18 2023 *)

my(x='x+O('x^30)); Vec(serlaplace(x^6*exp(x)*cosh(x)/6!)) \\ G. C. Greubel, Feb 05 2018

a(n) = (2^(n-1) + 0^n/2)*C(n+6,n).
a(n) = Sum_{j=0..n} C(n+6, j+6)*C(j+6, 6)*(1+(-1)^j)/2.
G.f.: (1 - 7*x + 42*x^2 - 140*x^3 + 280*x^4 - 336*x^5 + 224*x^6 - 64*x^7)/ (1-2*x)^7.
E.g.f.: (x^6/6!)*exp(x)*cosh(x) (with 6 leading zeros).
a(n) = ceiling(binomial(n+6,6)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 89/5 - 24*log(2).
Sum_{n>=0} (-1)^n/a(n) = 5832*log(3/2) - 11819/5. (End)

A082138 A transform of C(n,3).

Original entry on oeis.org

1, 4, 20, 80, 280, 896, 2688, 7680, 21120, 56320, 146432, 372736, 931840, 2293760, 5570560, 13369344, 31752192, 74711040, 174325760, 403701760, 928514048, 2122317824, 4823449600, 10905190400, 24536678400, 54962159616, 122607894528

Offset: 0

Paul Barry, Apr 06 2003

Fourth row of number array A082137. C(n,3) has e.g.f. (x^3/3!)exp(x). The transform averages the binomial and inverse binomial transforms.

			a(0) = (2^(-1) + 0^0/2)*C(3,0) = 2*(1/2) = 1 (using 0^0=1).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A080929, A082137, A082139, A082140, A082141, A080951, A057711.

a:=[4,20,80,280];; for n in [5..30] do a[n]:=8*a[n-1]-24*a[n-2] +32*a[n-3]-16*a[n-4]; od; Concatenation([1], a);

[(Ceiling(Binomial(n+3, 3)*2^(n-1))) : n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

[seq (ceil(binomial(n+3,3)*2^(n-1)),n=0..30)]; # Zerinvary Lajos, Nov 01 2006

Join[{1}, LinearRecurrence[{8,-24,32,-16}, {4,20,80,280}, 30]] (* G. C. Greubel, Jul 23 2019 *)

my(x='x+O('x^30)); Vec((1-4*x+12*x^2-16*x^3 + 8*x^4)/(1-2*x)^4) \\ G. C. Greubel, Jul 23 2019

((1-4*x+12*x^2-16*x^3+8*x^4)/(1-2*x)^4).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 23 2019

a(n) = (2^(n-1) + 0^n/2)*C(n+3, n).
a(n) = Sum_{j=0..n} C(n+3, j+3)*C(j+3, 3)*(1 + (-1)^j)/2.
G.f.: (1 - 4*x + 12*x^2 - 16*x^3 + 8*x^4)/(1-2*x)^4.
E.g.f.: (x^3/3!)*exp(x)*cosh(x) (preceded by 3 zeros).
a(n) = ceiling(binomial(n+3,3)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 12*log(2) - 7.
Sum_{n>=0} (-1)^n/a(n) = 108*log(3/2) - 43. (End)

cp's OEIS Frontend

A082143 First subdiagonal of number array A082137.

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A082144 A subdiagonal of number array A082137.

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A082145 A subdiagonal of number array A082137.

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A134308 A007318 * A082137.

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A069720 a(n) = 2^(n-1)*binomial(2*n-1, n).

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A069723 a(n) = 2^(n-1)*binomial(2*n-3, n-1).

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A134309 Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).

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A069720 a(n) = 2^(n-1)binomial(2n-1, n).

A069723 a(n) = 2^(n-1)binomial(2n-3, n-1).