A090181
Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0, 1, 55, 825, 4950, 13860
Offset: 0
Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1, 1;
[3] 0, 1, 3, 1;
[4] 0, 1, 6, 6, 1;
[5] 0, 1, 10, 20, 10, 1;
[6] 0, 1, 15, 50, 50, 15, 1;
[7] 0, 1, 21, 105, 175, 105, 21, 1;
[8] 0, 1, 28, 196, 490, 490, 196, 28, 1;
[9] 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1;
- Indranil Ghosh, Rows 0..100, flattened
- Yu Hin Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
- Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
- Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
- Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
- Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
- Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
- Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
- Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
- Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
- FindStat - Combinatorial Statistic Finder, The number of peaks of a Dyck path, The number of double rises of a Dyck path, The number of valleys of a Dyck path, The number of left oriented leafs except the first one of a binary tree, The number of left tunnels of a Dyck path
- Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Sum_{k=0..n} T(n,k)*x^k =
A000007(n),
A000108(n),
A006318(n),
A047891(n+1),
A082298(n),
A082301(n),
A082302(n),
A082305(n),
A082366(n),
A082367(n) for x=0,1,2,3,4,5,6,7,8,9. -
Philippe Deléham, Aug 10 2006
Sum_{k=0..n} x^(n-k)*T(n,k) =
A090192(n+1),
A000012(n),
A000108(n),
A001003(n),
A007564(n),
A059231(n),
A078009(n),
A078018(n),
A081178(n),
A082147(n),
A082181(n),
A082148(n),
A082173(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. -
Philippe Deléham, Oct 21 2006
Sum_{k=0..n} T(n,k)*x^k*(x-1)^(n-k) =
A000012(n),
A006318(n),
A103210(n),
A103211(n),
A133305(n),
A133306(n),
A133307(n),
A133308(n),
A133309(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. -
Philippe Deléham, Oct 20 2007
-
[[(&+[(-1)^(j-k)*Binomial(2*n-j,j)*Binomial(j,k)*Binomial(2*n-2*j,n-j)/(n-j+1): j in [0..n]]): k in [0..n]]: n in [0..10]];
-
A090181 := (n,k) -> binomial(n,n-k)*binomial(n-1,n-k)/(n-k+1):
seq(print( seq(A090181(n,k),k=0..n)),n=0..5); # Peter Luschny, May 10 2014
egf := 1+int((sqrt(t)*exp((1+t)*x)*BesselI(1,2*sqrt(t)*x))/x,x);
s := n -> n!*coeff(series(egf,x,n+2),x,n);
seq(print(seq(coeff(s(n),t,j),j=0..n)),n=0..9); # Peter Luschny, Oct 30 2014
T := proc(n, k) option remember; if k = n or k = 1 then 1 elif k < 1 then 0 else (2*n/k - 1) * T(n-1, k-1) + T(n-1, k) fi end:
for n from 0 to 8 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Dec 31 2024
-
Flatten[Table[Sum[(-1)^(j-k) * Binomial[2n-j,j] * Binomial[j,k] * CatalanNumber[n-j], {j, 0, n}], {n,0,11},{k,0,n}]] (* Indranil Ghosh, Mar 05 2017 *)
p[0, ] := 1; p[1, x] := x; p[n_, x_] := ((2 n - 1) (1 + x) p[n - 1, x] - (n - 2) (x - 1)^2 p[n - 2, x]) / (n + 1);
Table[CoefficientList[p[n, x], x], {n, 0, 9}] // TableForm (* Peter Luschny, Apr 26 2022 *)
-
c(n) = binomial(2*n,n)/ (n+1);
tabl(nn) = {for(n=0, nn, for(k=0, n, print1(sum(j=0, n, (-1)^(j-k) * binomial(2*n-j,j) * binomial(j,k) * c(n-j)),", ");); print(););};
tabl(11); \\ Indranil Ghosh, Mar 05 2017
-
from functools import cache
@cache
def Trow(n):
if n == 0: return [1]
if n == 1: return [0, 1]
if n == 2: return [0, 1, 1]
A = Trow(n - 2) + [0, 0]
B = Trow(n - 1) + [1]
for k in range(n - 1, 1, -1):
B[k] = (((B[k] + B[k - 1]) * (2 * n - 1)
- (A[k] - 2 * A[k - 1] + A[k - 2]) * (n - 2)) // (n + 1))
return B
for n in range(10): print(Trow(n)) # Peter Luschny, May 02 2022
-
def A090181_row(n):
U = [0]*(n+1)
for d in DyckWords(n):
U[d.number_of_peaks()] +=1
return U
for n in range(8): A090181_row(n) # Peter Luschny, May 10 2014
A132813
Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices.
Original entry on oeis.org
1, 1, 2, 1, 6, 3, 1, 12, 18, 4, 1, 20, 60, 40, 5, 1, 30, 150, 200, 75, 6, 1, 42, 315, 700, 525, 126, 7, 1, 56, 588, 1960, 2450, 1176, 196, 8, 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10
Offset: 0
First few rows of the triangle are:
1;
1, 2;
1, 6, 3;
1, 12, 18, 4;
1, 20, 60, 40, 5;
1, 30, 150, 200, 75, 6;
1, 42, 315, 700, 525, 126, 7;
...
- Reinhard Zumkeller, Rows n = 0..125 of table, flattened
- N. Alexeev and A. Tikhomirov, Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials, arXiv preprint arXiv:1501.04615 [math.PR], 2015.
- C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
- Robert. A. Sulanke, Counting Lattice Paths by Narayana Polynomials Electronic J. Combinatorics 7, No. 1, R40, 1-9, 2000.
-
Flat(List([0..10],n->List([0..n], k->(k+1)*Binomial(n+1,k+1)*Binomial(n+1,k)/(n+1)))); # Muniru A Asiru, Feb 26 2019
-
a132813 n k = a132813_tabl !! n !! k
a132813_row n = a132813_tabl !! n
a132813_tabl = zipWith (zipWith (*)) a007318_tabl $ tail a007318_tabl
-- Reinhard Zumkeller, Apr 04 2014
-
/* triangle */ [[(k+1)*Binomial(n+1,k+1)*Binomial(n+1,k)/(n+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 19 2014
-
P := (n, x) -> hypergeom([1-n, -n], [1], x): for n from 1 to 9 do PolynomialTools:-CoefficientList(simplify(P(n,x)),x) od; # Peter Luschny, Nov 26 2014
-
T[n_,k_]=Binomial[n-1,k-1]*Binomial[n,k-1]; Table[Table[T[n,k],{k,1,n}],{n,1,11}]; Flatten[%] (* Roger L. Bagula, Apr 09 2008 *)
P[n_, x_] := HypergeometricPFQ[{1-n, -n}, {1}, x]; Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 27 2014, after Peter Luschny *)
-
tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(binomial(n-1, k-1)*binomial(n, k-1) , ", ");););} \\ Michel Marcus, Feb 12 2014
-
def A132813(n,k): return binomial(n,k)*binomial(n+1,k)
print(flatten([[A132813(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 12 2025
A078009
a(0)=1, for n>=1 a(n) = Sum_{k=0..n} 5^k*N(n,k) where N(n,k) = C(n,k)*C(n,k+1)/n are the Narayana numbers (A001263).
Original entry on oeis.org
1, 1, 6, 41, 306, 2426, 20076, 171481, 1500666, 13386206, 121267476, 1112674026, 10318939956, 96572168916, 910896992856, 8650566601401, 82644968321226, 793753763514806, 7659535707782916, 74225795172589006, 722042370787826076
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
- Xiaomei Chen, Yuan Xiang, Counting generalized Schröder paths, arXiv:2009.04900 [math.CO], 2020.
-
R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+4*x - Sqrt(16*x^2-12*x+1))/(10*x) )); // G. C. Greubel, Jun 28 2019
-
[1] cat [&+[5^k*Binomial(n,k)*Binomial(n,k+1)/n:k in [0..n]]:n in [1..20]]; // Marius A. Burtea, Jan 21 2020
-
A078009_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+5*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A078009_list(20); # Peter Luschny, May 19 2011
-
Table[SeriesCoefficient[(1+4*x-Sqrt[16*x^2-12*x+1])/(10*x),{x,0,n}],{n,0,30}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 5];
Table[a[n], {n, 0, 30}] (* Peter Luschny, Mar 19 2018 *)
-
a(n)=sum(k=0,n,5^k/n*binomial(n,k)*binomial(n,k+1))
-
a=((1+4*x -sqrt(16*x^2-12*x+1))/(10*x)).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # G. C. Greubel, Jun 28 2019
A103365
First column of triangle A103364, which equals the matrix inverse of the Narayana triangle (A001263).
Original entry on oeis.org
1, -1, 2, -7, 39, -321, 3681, -56197, 1102571, -27036487, 810263398, -29139230033, 1238451463261, -61408179368043, 3513348386222286, -229724924077987509, 17023649385410772579, -1419220037471837658603, 132236541042728184852942, -13690229149108218523467549
Offset: 1
From _Paul D. Hanna_, Jan 31 2009: (Start)
G.f.: A(x) = 1 - x + 2*x^2/3 - 7*x^3/18 + 39*x^4/180 - 321*x^5/2700 +...
G.f.: A(x) = 1/B(x) where:
B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 + x^5/2700 +...+ x^n/[n!*(n+1)!/2^n] +... (End)
-
Table[(-1)^((n-1)/2) * (CoefficientList[Series[x/BesselJ[1,2*x],{x,0,40}],x])[[n]] * ((n+1)/2)! * ((n-1)/2)!,{n,1,41,2}] (* Vaclav Kotesovec, Mar 01 2014 *)
-
a(n)=if(n<1,0,(matrix(n,n,m,j,binomial(m-1,j-1)*binomial(m,j-1)/j)^-1)[n,1])
-
{a(n)=local(B=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(1/B,n)*n!*(n+1)!/2^n} \\ Paul D. Hanna, Jan 31 2009
A078018
a(n) = Sum_{k=0..n} 6^k*N(n,k), with a(0)=1, where N(n,k) = C(n,k) * C(n,k+1)/n are the Narayana numbers (A001263).
Original entry on oeis.org
1, 1, 7, 55, 469, 4237, 39907, 387739, 3858505, 39130777, 402972031, 4202705311, 44299426717, 471189693925, 5051001609115, 54513542257795, 591858123926545, 6459813793353265, 70837427884259575, 780073647992404615
Offset: 0
-
R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 + 5*x - Sqrt(25*x^2-14*x+1))/(12*x) )); // G. C. Greubel, Jun 29 2019
-
A078018_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+6*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A078018_list(19);
# Peter Luschny, May 19 2011
-
Table[SeriesCoefficient[(1+5*x-Sqrt[25*x^2-14*x+1])/(12*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[n_]:= Hypergeometric2F1[1 - n, -n, 2, 6]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Mar 19 2018 *)
-
a(n)=if(n<1,1,sum(k=0,n,6^k/n*binomial(n,k)*binomial(n,k+1)))
-
a=((1 + 5*x - sqrt(25*x^2-14*x+1))/(12*x)).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # G. C. Greubel, Jun 29 2019
A081178
a(0) = 1; for n>=1, a(n) = Sum_{k=0..n} 7^k*N(n,k), where N(n,k)=(1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
Original entry on oeis.org
1, 1, 8, 71, 680, 6882, 72528, 788019, 8766248, 99362894, 1143498224, 13326176998, 156950554384, 1865210341828, 22338852956064, 269355965364459, 3267146912972328, 39837475762660374, 488032452193307568
Offset: 0
-
B:=Binomial;
A081178:= func< n | n eq 0 select 1 else (&+[7^k*B(n,k)*B(n,k+1): k in [0..n]])/n >;
[A081178(n): n in [0..40]]; // G. C. Greubel, Jan 15 2024
-
A081178_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+7*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A081178_list(18); # Peter Luschny, May 19 2011
-
Table[SeriesCoefficient[(1+6*x-Sqrt[36*x^2-16*x+1])/(14*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 7];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)
-
a(n)=if(n<1,1,sum(k=0,n,7^k/n*binomial(n,k)*binomial(n,k+1)))
-
def A081178(n):
b=binomial;
if n==0: return 1
else: return (1/n)*sum(7^k*b(n,k)*b(n,k+1) for k in range(n+1))
[A081178(n) for n in range(41)] # G. C. Greubel, Jan 15 2024
A134291
Tenth column (and diagonal) of Narayana triangle A001263.
Original entry on oeis.org
1, 55, 1210, 15730, 143143, 1002001, 5725720, 27810640, 118195220, 449141836, 1551580888, 4936848280, 14620666060, 40648664980, 106847919376, 267119798440, 638337753625, 1464421905375, 3237143159250, 6917263803450
Offset: 0
- S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988.
Cf.
A134290 (ninth column of Narayana triangle).
-
List([0..30], n-> Binomial(n+10,10)*Binomial(n+9,8)/9); # G. C. Greubel, Aug 28 2019
-
[Binomial(n+10,10)*Binomial(n+9,8)/9: n in [0..30]]; // G. C. Greubel, Aug 28 2019
-
a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9))^2*(n+10))/ 1316818944000:
seq(a(n), n=0..19); # Peter Luschny, Sep 01 2016
-
Table[Binomial[n + 10, 10]*Binomial[n + 10, 9]/(n + 10), {n, 0, 30}] (* Wesley Ivan Hurt, Apr 25 2017 *)
-
vector(30, n, binomial(n+9,10)*binomial(n+8,8)/9) \\ G. C. Greubel, Aug 28 2019
-
[binomial(n+10,10)*binomial(n+9,8)/9 for n in (0..30)] # G. C. Greubel, Aug 28 2019
A082147
a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 8^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
Original entry on oeis.org
1, 1, 9, 89, 945, 10577, 123129, 1476841, 18130401, 226739489, 2878666857, 37006326777, 480750990993, 6301611631473, 83240669582937, 1106980509493641, 14808497812637121, 199138509770855489, 2690461489090104009
Offset: 0
-
a:=n->Sum([0..n],k->8^k*(1/n)*Binomial(n,k)*Binomial(n,k+1));;
Concatenation([1],List([1..18],n->a(n))); # Muniru A Asiru, Feb 10 2018
-
Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1+7*x-Sqrt(49*x^2-18*x+1))/(16*x))) // G. C. Greubel, Feb 05 2018
-
A082147_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+8*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A082147_list(18); # Peter Luschny, May 19 2011
-
Table[SeriesCoefficient[(1+7*x-Sqrt[49*x^2-18*x+1])/(16*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
f[n_] := Sum[ 8^k*Binomial[n, k]*Binomial[n, k + 1]/n, {k, 0, n}]; f[0] = 1; Array[f, 21, 0] (* Robert G. Wilson v, Feb 24 2018 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 8];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)
-
a(n)=if(n<1,1,sum(k=0,n,8^k/n*binomial(n,k)*binomial(n,k+1)))
A103364
Matrix inverse of the Narayana triangle A001263.
Original entry on oeis.org
1, -1, 1, 2, -3, 1, -7, 12, -6, 1, 39, -70, 40, -10, 1, -321, 585, -350, 100, -15, 1, 3681, -6741, 4095, -1225, 210, -21, 1, -56197, 103068, -62916, 19110, -3430, 392, -28, 1, 1102571, -2023092, 1236816, -377496, 68796, -8232, 672, -36, 1, -27036487, 49615695, -30346380, 9276120, -1698732, 206388
Offset: 1
Rows begin:
1;
-1, 1;
2, -3, 1;
-7, 12, -6, 1;
39, -70, 40, -10, 1;
-321, 585, -350, 100, -15, 1;
3681, -6741, 4095, -1225, 210, -21, 1;
-56197, 103068, -62916, 19110, -3430, 392, -28, 1;
1102571, -2023092, 1236816, -377496, 68796, -8232, 672, -36, 1;
...
From _Peter Bala_, Aug 09 2013: (Start)
The real zeros of the row polynomials R(n,x) appear to converge to zeros of E(alpha*x) as n increases, where alpha = -3.67049 26605 ... ( = -(A115369/2)^2).
Polynomial | Real zeros to 5 decimal places
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
R(5,x) | 1, 3.57754, 3.81904
R(10,x) | 1, 3.35230, 7.07532, 9.14395
R(15,x) | 1, 3.35231, 7.04943, 12.09668, 15.96334
R(20,x) | 1, 3.35231, 7.04943, 12.09107, 18.47845, 24.35255
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Function |
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
E(alpha*x) | 1, 3.35231, 7.04943, 12.09107, 18.47720, 26.20778, ...
Note: The n-th zero of E(alpha*x) may be calculated in Maple 17 using the instruction evalf( BesselJZeros(1,n)/ BesselJZeros(1,1))^2 ). (End)
-
T[n_, 1]:= Last[Table[(-1)^(n - 1)*(CoefficientList[Series[x/BesselJ[1, 2*x], {x, 0, 1000}], x])[[k]]*(n)!*(n - 1)!, {k, 1, 2*n - 1, 2}]]
T[n_, n_] := 1; T[2, 1] := -1; T[3, 1] := 2; T[n_, k_] := T[n, k] = T[n - 1, k - 1]*n*(n - 1)/(k*(k - 1)); Table[T[n, k], {n, 1, 50}, {k, 1, n}] // Flatten (* G. C. Greubel, Jan 04 2016 *)
T[n_, n_] := 1; T[n_, 1]/;n>1 := T[n, 1] = -Sum[T[n, j], {j, 2, n}]; T[n_, k_]/;1Oliver Seipel, Jan 01 2025 *)
-
T(n,k)=if(n
A134289
Eighth column (and diagonal) of Narayana triangle A001263.
Original entry on oeis.org
1, 36, 540, 4950, 32670, 169884, 736164, 2760615, 9202050, 27810640, 77364144, 200443464, 488259720, 1126753200, 2478857040, 5226256926, 10606227291, 20796524100, 39525557500, 73018266750, 131432880150, 231003243900, 397179490500, 669161098125, 1106346348900
Offset: 0
- S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988.
- T. D. Noe, Table of n, a(n) for n = 0..1000
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 25.
- Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).
Cf.
A134288 (seventh column of Narayana triangle).
Cf.
A134290 (ninth column of Narayana triangle).
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List([0..30], n-> Binomial(n+8,8)*Binomial(n+7,6)/7); # G. C. Greubel, Aug 28 2019
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[Binomial(n+8,8)*Binomial(n+7,6)/7: n in [0..30]]; // G. C. Greubel, Aug 28 2019
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a := n -> ((n+1)*((n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7))^2*(n+8))/203212800;
seq(a(n), n=0..24); # Peter Luschny, Sep 01 2016
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Table[(Binomial[n + 8, 8] Binomial[n + 8, 7])/(n + 8), {n, 0, 30}] (* or *) LinearRecurrence[{15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1},{1, 36, 540, 4950, 32670, 169884, 736164, 2760615, 9202050, 27810640, 77364144, 200443464, 488259720, 1126753200, 2478857040}, 30] (* Harvey P. Dale, Jul 23 2012 *)
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vector(30, n, binomial(n+7,8)*binomial(n+6,6)/7) \\ G. C. Greubel, Aug 28 2019
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[binomial(n+8,8)*binomial(n+7,6)/7 for n in (0..30)] # G. C. Greubel, Aug 28 2019
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