A115129
Partial sums of A005587. Fourth column of triangle A115127.
Original entry on oeis.org
14, 56, 146, 311, 586, 1015, 1652, 2562, 3822, 5522, 7766, 10673, 14378, 19033, 24808, 31892, 40494, 50844, 63194, 77819, 95018, 115115, 138460, 165430, 196430, 231894, 272286, 318101, 369866, 428141, 493520, 566632, 648142, 738752, 839202
Offset: 0
-
LinearRecurrence[{6,-15,20,-15,6,-1},{14,56,146,311,586,1015},40] (* or *) CoefficientList[Series[(14-28x+20x^2-5x^3)/(1-x)^6,{x,0,40}],x] (* Harvey P. Dale, Apr 24 2016 *)
A106566
Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 14, 14, 9, 4, 1, 0, 42, 42, 28, 14, 5, 1, 0, 132, 132, 90, 48, 20, 6, 1, 0, 429, 429, 297, 165, 75, 27, 7, 1, 0, 1430, 1430, 1001, 572, 275, 110, 35, 8, 1, 0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1
Offset: 0
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 2, 2, 1;
0, 5, 5, 3, 1;
0, 14, 14, 9, 4, 1;
0, 42, 42, 28, 14, 5, 1;
0, 132, 132, 90, 48, 20, 6, 1;
From _Paul Barry_, Sep 28 2009: (Start)
Production array is
0, 1,
0, 1, 1,
0, 1, 1, 1,
0, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)
- Alois P. Heinz, Rows n = 0..140, flattened
- Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
- Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
- Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
- Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
- F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
- D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
- E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
- FindStat - Combinatorial Statistic Finder, The number of touch points of a Dyck path, The number of initial rises of a Dyck paths, The number of nodes on the left branch of the tree, The number of subtrees.
- R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
- A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
- L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.
Column k for k = 0, 1, 2, ..., 13:
A000007,
A000108,
A000108,
A000245,
A002057,
A000344,
A003517,
A000588,
A003517,
A001392,
A003518,
A000589,
A003519,
A000590Generalized Catalan numbers C(x, n) for -11 <= x <= 10:
A064333,
A064332,
A064331,
A064330,
A064329,
A064328,
A064327,
A064326,
A064325,
A064311,
A064310,
A000012,
A000108,
A064062,
A064063,
A064087,
A064088,
A064089,
A064090,
A064091,
A064092,
A064093.
-
A106566:= func< n,k | n eq 0 select 1 else (k/n)*Binomial(2*n-k-1, n-k) >;
[A106566(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 06 2021
-
A106566 := proc(n,k)
if n = 0 then
1;
elif k < 0 or k > n then
0;
else
binomial(2*n-k-1,n-k)*k/n ;
end if;
end proc: # R. J. Mathar, Mar 01 2015
-
T[n_, k_] := Binomial[2n-k-1, n-k]*k/n; T[0, 0] = 1; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2017 *)
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, #(1-Sqrt[1-4#])/(2#)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)
-
{T(n, k) = if( k<=0 || k>n, n==0 && k==0, binomial(2*n - k, n) * k/(2*n - k))}; /* Michael Somos, Oct 01 2022 */
-
def A106566(n, k): return 1 if (n==0) else (k/n)*binomial(2*n-k-1, n-k)
flatten([[A106566(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 06 2021
A214292
Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n with T(n,0) = n and T(n,n) = -n.
Original entry on oeis.org
0, 1, -1, 2, 0, -2, 3, 2, -2, -3, 4, 5, 0, -5, -4, 5, 9, 5, -5, -9, -5, 6, 14, 14, 0, -14, -14, -6, 7, 20, 28, 14, -14, -28, -20, -7, 8, 27, 48, 42, 0, -42, -48, -27, -8, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9, 10, 44, 110, 165, 132, 0, -132, -165, -110, -44, -10
Offset: 0
The triangle begins:
0: 0
1: 1 -1
2: 2 0 -2
3: 3 2 -2 -3
4: 4 5 0 -5 -4
5: 5 9 5 -5 -9 -5
6: 6 14 14 0 -14 -14 -6
7: 7 20 28 14 -14 -28 -20 -7
8: 8 27 48 42 0 -42 -48 -27 -8
9: 9 35 75 90 42 -42 -90 -75 -35 -9
10: 10 44 110 165 132 0 -132 -165 -110 -44 -10
11: 11 54 154 275 297 132 -132 -297 -275 -154 -54 -11 .
Cf.
A007318,
A000004,
A000096,
A000108,
A000245,
A000344,
A000588,
A000589,
A000590,
A001392,
A002057,
A003517,
A003518,
A003519,
A005557,
A005586,
A005587,
A008313,
A014495,
A064059,
A064061,
A080956,
A090749,
A097808,
A112467,
A124087,
A124088,
A129936,
A259525.
-
a214292 n k = a214292_tabl !! n !! k
a214292_row n = a214292_tabl !! n
a214292_tabl = map diff $ tail a007318_tabl
where diff row = zipWith (-) (tail row) row
-
row[n_] := Table[Binomial[n, k], {k, 0, n}] // Differences;
T[n_, k_] := row[n + 1][[k + 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 31 2018 *)
A120730
Another version of Catalan triangle A009766.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 5, 4, 1, 0, 0, 0, 5, 9, 5, 1, 0, 0, 0, 0, 14, 14, 6, 1, 0, 0, 0, 0, 14, 28, 20, 7, 1, 0, 0, 0, 0, 0, 42, 48, 27, 8, 1, 0, 0, 0, 0, 0, 42, 90, 75, 35, 9, 1, 0, 0, 0, 0, 0, 0, 132, 165, 110, 44, 10, 1
Offset: 0
As a triangle, this begins:
1;
0, 1;
0, 1, 1;
0, 0, 2, 1;
0, 0, 2, 3, 1;
0, 0, 0, 5, 4, 1;
0, 0, 0, 5, 9, 5, 1;
0, 0, 0, 0, 14, 14, 6, 1;
...
-
A120730:= func< n,k | n gt 2*k select 0 else Binomial(n, k)*(2*k-n+1)/(k+1) >;
[A120730(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Nov 07 2022
-
G := 4*z/((2*z-1+sqrt(1-4*z^2*t))*(1+sqrt(1-4*z^2*t))): Gser := simplify(series(G, z = 0, 13)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form # Emeric Deutsch, Jun 19 2011
# second Maple program:
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
end:
T:= (n, k)-> b(n, 2*k-n):
seq(seq(T(n, k), k=0..n), n=0..14); # Alois P. Heinz, Oct 13 2022
-
b[x_, y_]:= b[x, y]= If[y<0 || y>x, 0, If[x==0, 1, Sum[b[x-1, y+j], {j, {-1, 1}}] ]];
T[n_, k_] := b[n, 2 k - n];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Oct 21 2022, after Alois P. Heinz *)
T[n_, k_]:= If[n>2*k, 0, Binomial[n, k]*(2*k-n+1)/(k+1)];
Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 07 2022 *)
-
def A120730(n,k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
flatten([[A120730(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Nov 07 2022
A005582
a(n) = n*(n+1)*(n+2)*(n+7)/24.
Original entry on oeis.org
0, 2, 9, 25, 55, 105, 182, 294, 450, 660, 935, 1287, 1729, 2275, 2940, 3740, 4692, 5814, 7125, 8645, 10395, 12397, 14674, 17250, 20150, 23400, 27027, 31059, 35525, 40455, 45880, 51832, 58344, 65450, 73185, 81585, 90687, 100529, 111150, 122590, 134890
Offset: 0
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
- Vladimir S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19. [From Vladimir Shevelev, Apr 12 2010]
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the case k=4) (First published: San Francisco: Holden-Day, Inc., 1964)
- Vincenzo Librandi, Table of n, a(n) for n = 0..10000
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988
- F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
- Milan Janjic, Two Enumerative Functions
- P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380. [See p. 301]
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.
- Index entries for sequences related to Chebyshev polynomials.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
-
[seq(binomial(n,4)+2*binomial(n,3), n=2..43)]; # Zerinvary Lajos, Jul 26 2006
seq((n+4)*binomial(n,4)/n, n=3..43); # Zerinvary Lajos, Feb 28 2007
A005582:=(-2+z)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation
-
Table[n(n+1)(n+2)(n+7)/24,{n,0,40}] (* Harvey P. Dale, Jun 01 2012 *)
-
concat(0, Vec(x*(2-x)/(1-x)^5 + O(x^100))) \\ Altug Alkan, Dec 10 2015
More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000
A064059
Seventh column of Catalan triangle A009766.
Original entry on oeis.org
132, 429, 1001, 2002, 3640, 6188, 9996, 15504, 23256, 33915, 48279, 67298, 92092, 123970, 164450, 215280, 278460, 356265, 451269, 566370, 704816, 870232, 1066648, 1298528, 1570800, 1888887, 2258739, 2686866, 3180372, 3746990, 4395118, 5133856, 5973044
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
-
A064059:= func< n | (n+1)*Binomial(n+12,5)/6 >;
[A064059(n): n in [0..40]]; // G. C. Greubel, Sep 27 2024
-
[seq(binomial(n+1,6)-2*binomial(n,5),n=12..55)]; # Zerinvary Lajos, Jul 19 2006
-
CoefficientList[Series[(42 z^5-252 z^4+616 z^3-770 z^2+495 z-132)/(z-1)^7, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{132,429,1001,2002,3640,6188,9996},40] (* Harvey P. Dale, Jan 08 2025 *)
-
def A064059(n): return (n+1)*binomial(n+12,5)//6
[A064059(n) for n in range(41)] # G. C. Greubel, Sep 27 2024
A192174
Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.
Original entry on oeis.org
-1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, 0, -2, 0, -1, 0, 1, 0, -3, 0, 0, 0, 1, 1, 0, -4, 0, 2, 0, 2, 0, 1, 0, -5, 0, 5, 0, 2, 0, -1, 1, 0, -6, 0, 9, 0, 0, 0, -3, 0, 1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1, 1, 0, -8, 0, 20, 0, -14, 0, -5, 0, 4, 0
Offset: 0
Triangle begins
-1; # -1
1, 0; # x
1, 0, 1; # x^2+1
1, 0, 0, 0; # x^3
1, 0, -1, 0, -1; # x^4-x^2-1
1, 0, -2, 0, -1, 0;
1, 0, -3, 0, 0, 0, 1;
1, 0, -4, 0, 2, 0, 2, 0;
1, 0, -5, 0, 5, 0, 2, 0, -1;
1, 0, -6, 0, 9, 0, 0, 0, -3, 0;
1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1;
1, 0, -8, 0, 20, 0,-14, 0, -5, 0, 4, 0;
1, 0, -9, 0, 27, 0,-28, 0, 0, 0, 9, 0, -1;
-
p:= proc(n,x) option remember: if n=0 then -1 elif n=1 then x elif n>=2 then x*procname(n-1,x)-procname(n-2,x) fi: end: A192174 := proc(n,k): coeff(p(n,x),x,n-k): end: seq(seq(A192174(n,k),k=0..n), n=0..11); # Johannes W. Meijer, Aug 21 2011
A176239
Shifted signed Catalan triangle T(n,k) = (-1)^*(n+k+1)*A009766(n,k-n+1) read by rows.
Original entry on oeis.org
0, -1, 1, -1, 0, 2, 0, 1, -2, 2, 0, -5, 0, 0, 1, -3, 5, -5, 0, 14, 0, 0, 0, 1, -4, 9, -14, 14, 0, -42, 0, 0, 0, 0, 1, -5, 14, -28, 42, -42, 0, 132, 0, 0, 0, 0, 0, 1, -6, 20, -48, 90, -132, 132, 0, -429, 0, 0, 0, 0, 0, 0, 1, -7, 27, -75, 165, -297, 429, -429, 0, 1430
Offset: 0
The triangle starts in row n=0 with columns 0 <= k < 2*(n+1) as:
0,-1; (-1)^k*k A001477
1,-1,.0,.2; (-1)^(k+1)*(k+1)*(k-2)/2 A080956, A000096
0,.1,-2,.2,.0,-5; (-1)^n*k*(k+1)*(k-4)/6 A129936, A005586
0,.0,.1,-3,.5,-5,..0,.14; (-1)^k*k*(k+1)*(k-1)*(k-6)/24, A005587
0,.0,.0,.1,-4,.9,-14,.14,.0,-42; A005557, A034807
0,.0,.0,.0,.1,-5,.14,-28,42,-42,0,132;
-
A009766 := proc(n,k) if k<0 or k >n then 0; else binomial(n+k,n)*(n-k+1)/(n+1) ; end if; end proc:
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A176239 := proc(n,k) if k <= 2*n-1 then (-1)^(n+k+1)*A009766(n,k-n+1) elif k = 2*n then 0; elif k < 2*(n+1) then (-1)^(n+1)*A000108(n+1); else 0; end if; end proc: # R. J. Mathar, Dec 03 2010
A115126
First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).
Original entry on oeis.org
1, 2, 2, 3, 5, 5, 4, 9, 14, 14, 5, 14, 28, 42, 42, 6, 20, 48, 90, 132, 132, 7, 27, 75, 165, 297, 429, 429, 8, 35, 110, 275, 572, 1001, 1430, 1430, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796, 16796, 11, 65, 273, 910
Offset: 1
[1];[2,2];[3,5,5];[4,9,14,14];...
a(4,2) = 9 = binomial(6,2)*3/5.
- B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
- B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.
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