A014532 -id:A014532 - cp's OEIS frontend

A111808 Left half of trinomial triangle (A027907), triangle read by rows.

1, 1, 1, 1, 2, 3, 1, 3, 6, 7, 1, 4, 10, 16, 19, 1, 5, 15, 30, 45, 51, 1, 6, 21, 50, 90, 126, 141, 1, 7, 28, 77, 161, 266, 357, 393, 1, 8, 36, 112, 266, 504, 784, 1016, 1107, 1, 9, 45, 156, 414, 882, 1554, 2304, 2907, 3139, 1, 10, 55, 210, 615, 1452, 2850, 4740, 6765, 8350

Offset: 1

Reinhard Zumkeller, Aug 17 2005

Consider a doubly infinite chessboard with squares labeled (n,k), ranks or rows n in Z, files or columns k in Z (Z denotes ...,-2,-1,0,1,2,... ); number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,n-k). - Harrie Grondijs, May 27 2005. Cf. A026300, A114929, A114972.

Triangle of numbers C^(2)(n-1,k), n>=1, of combinations with repetitions from elements {1,2,...,n} over k, such that every element i, i=1,...,n, appears in a k-combination either 0 or 1 or 2 times (cf. also A213742-A213745). - Vladimir Shevelev and Peter J. C. Moses, Jun 19 2012

Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Trinomial Triangle
Eric Weisstein's World of Mathematics, Trinomial Coefficient

Row sums give A027914; central terms give A027908;
T(n, 0) = 0;
T(n, 1) = n for n>1;
T(n, 2) = A000217(n) for n>1;
T(n, 3) = A005581(n) for n>2;
T(n, 4) = A005712(n) for n>3;
T(n, 5) = A000574(n) for n>4;
T(n, 6) = A005714(n) for n>5;
T(n, 7) = A005715(n) for n>6;
T(n, 8) = A005716(n) for n>7;
T(n, 9) = A064054(n-5) for n>8;
T(n, n-5) = A098470(n) for n>4;
T(n, n-4) = A014533(n-3) for n>3;
T(n, n-3) = A014532(n-2) for n>2;
T(n, n-2) = A014531(n-1) for n>1;
T(n, n-1) = A005717(n) for n>0;
T(n, n) = central terms of A027907 = A002426(n).

T := (n,k) -> simplify(GegenbauerC(k, -n, -1/2)):
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, May 09 2016

Table[GegenbauerC[k, -n, -1/2], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

(1 + x + x^2)^n = Sum(T(n,k)*x^k: 0<=k<=n) + Sum(T(n,k)*x^(2*n-k): 0<=k

T(n, k) = A027907(n, k) = Sum_{i=0,..,(k/2)} binomial(n, n-k+2*i) * binomial(n-k+2*i, i), 0<=k<=n.

T(n, k) = GegenbauerC(k, -n, -1/2). - Peter Luschny, May 09 2016

Corrected and edited by Johannes W. Meijer, Oct 05 2010

A243827 Number A(n,k) of Dyck paths of semilength n having exactly one occurrence of the consecutive step pattern given by the binary expansion of k, where 1=U=(1,1) and 0=D=(1,-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 4, 6, 1, 0, 0, 0, 0, 1, 2, 11, 10, 1, 0, 0, 0, 0, 0, 4, 6, 26, 15, 1, 0, 0, 0, 0, 0, 1, 11, 16, 57, 21, 1, 0, 0, 0, 0, 0, 1, 4, 26, 45, 120, 28, 1, 0, 0, 0, 0, 1, 1, 5, 15, 57, 126, 247, 36, 1, 0, 0

Offset: 0

Alois P. Heinz, Jun 11 2014

			Square array A(n,k) begins:
  0, 0, 0,  0,   0,    0,   0,    0,    0,    0, ...
  1, 1, 1,  0,   0,    0,   0,    0,    0,    0, ...
  0, 0, 1,  1,   1,    1,   1,    0,    0,    0, ...
  0, 0, 1,  3,   4,    2,   4,    1,    1,    1, ...
  0, 0, 1,  6,  11,    6,  11,    4,    5,    5, ...
  0, 0, 1, 10,  26,   16,  26,   15,   21,   17, ...
  0, 0, 1, 15,  57,   45,  57,   50,   78,   54, ...
  0, 0, 1, 21, 120,  126, 120,  161,  274,  177, ...
  0, 0, 1, 28, 247,  357, 247,  504,  927,  594, ...
  0, 0, 1, 36, 502, 1016, 502, 1554, 3061, 1997, ...

Alois P. Heinz, antidiagonals n = 0..140, flattened

Columns k=2-10 give: A000012(n) for n>0, A000217(n-1) for n>0, A000295(n-1) for n>0, A005717(n-1) for n>1, A000295(n-1) for n>0, A014532(n-2) for n>2, A108863, A244235, A244236.
Main diagonal gives A243770 or column k=1 of A243752.
Cf. A243753, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.

A014531 Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 2nd column from the center.

Original entry on oeis.org

1, 3, 10, 30, 90, 266, 784, 2304, 6765, 19855, 58278, 171106, 502593, 1477035, 4343160, 12778152, 37616427, 110797569, 326527350, 962803170, 2840372304, 8383467708, 24755608584, 73133433800, 216143407675, 639062383401

Offset: 1

N. J. A. Sloane

Number of "up" steps in all Motzkin paths of length n+1. E.g. a(2)=3 because in the four Motzkin paths of length 3, HHH, HUD, UDH and UHD, where H=(1,0), U=(1,1), D=(1,-1), we have altogether three U steps. - Emeric Deutsch, Dec 26 2003
a(n-1) = A111808(n,n-2) for n>1. - Reinhard Zumkeller, Aug 17 2005
a(n) = number of paths in the half-plane x>=0, from (0,0) to (n+1,2), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=2, we have the 3 paths: UUH, HUU, UHU. - José Luis Ramírez Ramírez, Apr 19 2015

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.

G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 21-22.
Mark Shattuck, Subword Patterns in Smooth Words, Enum. Comb. Appl. (2024) Vol. 4, No. 4, Art. No. S2R32. See p. 6.
Eric Weisstein's World of Mathematics, Trinomial Coefficient.

Cf. A027907, A005717, A055151.

First differences are in A025180.

seq( add(binomial(i+1,k)*binomial(i-k+1,k+2), k=0..floor(i/2)), i=1..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
a := n -> simplify(GegenbauerC(n-1, -n-1, -1/2)):
seq(a(n), n=1..26); # Peter Luschny, May 09 2016

Table[Sum[Binomial[i + 1, k]*Binomial[i - k + 1, k + 2], {k, 0, Floor[i/2]}], {i, 30}] (* Michael De Vlieger, Apr 20 2015 *)
Table[GegenbauerC[n - 1, -n - 1, -1/2], {n,1,50}] (* G. C. Greubel, Feb 28 2017 *)

for(n=1,25, print1(sum(k=0,n+1, binomial(n+1,k)*binomial(n-k+1,k+2)), ", ")) \\ G. C. Greubel, Feb 28 2017

a = lambda n: n*(n+1)*hypergeometric([(1-n)/2, 1-n/2], [3], 4)/2
[simplify(a(n)) for n in (1..26)] # Peter Luschny, Nov 23 2014

a(n) = A002426(n+1)-A001006(n+1) = a(n-1)+A005717(n)+A014532(n-2) - Henry Bottomley, May 15 2001
E.g.f.: exp(x)*(2*x*BesselI(1, 2*x)+(x-2)*BesselI(2, 2*x))/x. - Vladeta Jovovic, Aug 21 2003
G.f.: [1-2z-z^2-(1-z)q]/(2z^3q), where q=sqrt(1-2z-3z^2). - Emeric Deutsch, Dec 26 2003
a(n) = Sum_{k=0..n+1} C(n+1,k)*C(n-k+1,k+2). - Paul Barry, Sep 20 2004
D-finite with recurrence (n+3)*(n-1)*a(n) -(n+1)*(2n+1)*a(n-2)-3*n*(n+1)*a(n-2)=0. - R. J. Mathar, Dec 08 2011
a(n) = n*(n+1)*hypergeom([(1-n)/2, 1-n/2], [3], 4)/2. - Peter Luschny, Nov 23 2014
G.f.: z*M(z)^2/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) = GegenbauerC(n-1, -n-1, -1/2). - Peter Luschny, May 09 2016
a(n) = Sum_{k>0} k * A055151(n+1,k). - Alois P. Heinz, Mar 29 2020

More terms from James Sellers, Feb 05 2000

A025181 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 3. Also a(n) = T(n,n-3), where T is the array defined in A025177.

Original entry on oeis.org

1, 3, 11, 35, 111, 343, 1050, 3186, 9615, 28897, 86592, 258908, 772863, 2304225, 6863496, 20429784, 60779403, 180751617, 537386595, 1597372371, 4747537641, 14108988509, 41928203694, 124598731750, 370279082745, 1100428538391, 3270534249843

Offset: 3

Clark Kimberling

nonn

Cf. A025568.

First differences of A014532. First differences are in A026070.

Conjecture: +(n+3)*a(n) +(-5*n-7)*a(n-1) +(3*n-7)*a(n-2) +(11*n-7)*a(n-3) +4*(-n+6)*a(n-4) +6*(-n+5)*a(n-5)=0. - R. J. Mathar, Feb 25 2015

Conjecture: -(n+3)*(n-3)*(4*n^2-12*n+17)*a(n) +(n-1)*(8*n^3-20*n^2+30*n-81)*a(n-1) +3*(n-1)*(n-2)*(4*n^2-4*n+9)*a(n-2)=0. - R. J. Mathar, Feb 25 2015

A092107 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UUU's (triple rises) where U=(1,1). Rows have 1,1,1,2,3,4,5,... entries, respectively.

Original entry on oeis.org

1, 1, 2, 4, 1, 9, 4, 1, 21, 15, 5, 1, 51, 50, 24, 6, 1, 127, 161, 98, 35, 7, 1, 323, 504, 378, 168, 48, 8, 1, 835, 1554, 1386, 750, 264, 63, 9, 1, 2188, 4740, 4920, 3132, 1335, 390, 80, 10, 1, 5798, 14355, 17028, 12507, 6237, 2200, 550, 99, 11, 1, 15511, 43252, 57816

Offset: 0

Emeric Deutsch, Mar 29 2004

Column 0 gives the Motzkin numbers (A001006), column 1 gives A014532. Row sums are the Catalan numbers (A000108).

Equal to A171380*B (without the zeros), B = A007318. - Philippe Deléham, Dec 10 2009

			T(5,2) = 5 because we have (U[UU)U]DUDDDD, (U[UU)U]DDUDDD, (U[UU)U]DDDUDD, (U[UU)U]DDDDUD and UD(U[UU)U]DDDD, where U=(1,1), D=(1,-1); the triple rises are shown between parentheses.
[1],[1],[2],[4, 1],[9, 4, 1],[21, 15, 5, 1],[51, 50, 24, 6, 1],[127, 161, 98, 35, 7, 1]
Triangle starts:
     1;
     1;
     2;
     4,    1;
     9,    4,    1;
    21,   15,    5,    1;
    51,   50,   24,    6,    1;
   127,  161,   98,   35,    7,    1;
   323,  504,  378,  168,   48,    8,    1;
   835, 1554, 1386,  750,  264,   63,    9,    1;
  2188, 4740, 4920, 3132, 1335,  390,   80,   10,    1;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 14.
Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, and Teresa Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv:1812.07112 [math.CO], 2018.
FindStat - Combinatorial Statistic Finder, The number of occurrences of the contiguous pattern [.,[.,[.,.]]].
Lara Pudwell, On the distribution of peaks (and other statistics), 16th International Conference on Permutation Patterns, Dartmouth College, 2018.
Toufik Mansour and Mark Shattuck, Counting occurrences of subword patterns in non-crossing partitions, Art Disc. Appl. Math. (2022).
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, and Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2.
Yidong Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 287 (2004), 177-186.

Cf. A000108, A001006, A001405, A007318, A014532, A033321, A171380, A243752, A243753.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, expand(b(x-1, y-1, min(t+1,2))*
      `if`(t=2, z, 1) +b(x-1, y+1, 0))))
    end:
T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 0)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 11 2014

b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, Expand[b[x-1, y-1, Min[t+1, 2]]*If[t == 2, z, 1] + b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Alois P. Heinz *)

G.f.: G(t, z) satisfies z(t+z-tz)G^2 - (1-z+tz)G + 1 = 0.

Sum_{k=0..n} T(n,k)*x^k = A001405(n), A001006(n), A000108(n), A033321(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 10 2009

A025568 a(n) = T(n,n+2) where T is the array defined in A025564.

Original entry on oeis.org

1, 5, 19, 65, 211, 665, 2058, 6294, 19095, 57607, 173096, 518596, 1550367, 4627455, 13795176, 41088456, 122297643, 363828663, 1081966875, 3216725841, 9561635853, 28418162003, 84455354206, 250982289650, 745860104145, 2216567725281

Offset: 1

Clark Kimberling

nonn

First differences are pairwise sums of A025181.

Pairwise sums of A014532.

A026070 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 3. Also a(n) = T(n,n-3), where T is the array defined in A024996.

Original entry on oeis.org

1, 2, 8, 24, 76, 232, 707, 2136, 6429, 19282, 57695, 172316, 513955, 1531362, 4559271, 13566288, 40349619, 119972214, 356634978, 1059985776, 3150165270, 9361450868, 27819215185, 82670528056, 245680350995, 730149455646, 2170105711452

Offset: 3

Clark Kimberling

nonn

First differences of A025181. Second differences of A014532.

Conjecture: (n+3)*a(n) +(-5*n-6)*a(n-1) +(3*n-11)*a(n-2) +(11*n-8)*a(n-3) +2*(-2*n+17)*a(n-4) +6*(-n+6)*a(n-5)=0. - R. J. Mathar, Jun 23 2013

A098470 Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 5th column from the center.

Original entry on oeis.org

1, 6, 28, 112, 414, 1452, 4917, 16236, 52624, 168168, 531531, 1665456, 5182008, 16031952, 49366674, 151419816, 462919401, 1411306358, 4292487562, 13029127584, 39478598170, 119439969220, 360881425710, 1089126806040

Offset: 5

Eric W. Weisstein, Sep 09 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 5..1005
Eric Weisstein's World of Mathematics, Trinomial Coefficient

Cf. A002426, A005717, A014531, A014532, A014533.

# Assuming offset 0:
a := n -> simplify(GegenbauerC(n, -n-5, -1/2)):
seq(a(n), n=0..25); # Peter Luschny, May 09 2016

Table[GegenbauerC[n, -n - 5, -1/2], {n,0,50}] (* G. C. Greubel, Feb 28 2017 *)

x='x + O('x^50); Vec(32*x^5/(sqrt((1+x)*(1-3*x))*(1-x-sqrt((1+x)*(1-3*x)))^5)) \\ G. C. Greubel, Feb 28 2017

(n^2-25)*a(n) = n*(2*n-1)*a(n-1) + 3*n*(n-1)*a(n-2). - Vladeta Jovovic, Sep 18 2004
G.f.: 32*x^5/(sqrt((1+x)*(1-3*x))*(1-x-sqrt((1+x)*(1-3*x)))^5). - Vladeta Jovovic, Sep 18 2004
a(n) = A111808(n,n-5). - Reinhard Zumkeller, Aug 17 2005
Assuming offset 0: a(n) = GegenbauerC(n,-n-5,-1/2) and a(n) = binomial(10+2*n,n)* hypergeom([-n, -n-10], [-9/2-n], 1/4). - Peter Luschny, May 09 2016
a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 09 2021

A116401 Triangle whose k-th column has e.g.f. exp(x)*sum{j=0..k, Bessel_I(k+j,2x)}.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 7, 9, 4, 1, 19, 26, 15, 5, 1, 51, 75, 50, 21, 6, 1, 141, 216, 161, 78, 28, 7, 1, 393, 623, 504, 273, 113, 36, 8, 1, 1107, 1800, 1554, 918, 423, 157, 45, 9, 1, 3139, 5211, 4740, 3006, 1506, 625, 211, 55, 10, 1, 8953, 15115, 14355, 9657, 5182, 2343, 891, 276

Offset: 0

Paul Barry, Feb 13 2006

First column is A002426. Second column is A005774. Third column is A014532.

			Triangle begins:
  1,
  1, 1,
  3, 3, 1,
  7, 9, 4, 1,
  19, 26, 15, 5, 1,
  51, 75, 50, 21, 6, 1
  ...

Cf. A002426, A005774, A014532.

A126218 Triangle read by rows: T(n,k) is the number of 0-1-2 trees (i.e., ordered trees with all vertices of outdegree at most two) with n edges and k pairs of adjacent vertices of outdegree 2.

Original entry on oeis.org

1, 1, 2, 4, 7, 2, 13, 8, 26, 20, 5, 52, 50, 25, 104, 130, 75, 14, 212, 322, 217, 84, 438, 770, 644, 294, 42, 910, 1836, 1806, 952, 294, 1903, 4362, 4830, 3108, 1176, 132, 4009, 10268, 12738, 9576, 4188, 1056, 8494, 24032, 33219, 27948, 14760, 4752, 429, 18080

Offset: 0

Emeric Deutsch, Dec 24 2006

Row n has floor(n/2) terms (n >= 2).
Row sums are the Motzkin numbers (A001006).
T(n,1) = A023431(n+1).
Sum_{k=0..floor(n/2)-1} k*T(n,k) = 2*A014532(n-3) (n >= 4).

			Triangle starts:
   1;
   1;
   2;
   4;
   7,  2;
  13,  8;
  26, 20,  5;
  52, 50, 25;

Cf. A001006, A014532, A023431.

G:=1/2*(2*z^2*t^2-z+4*z^3*t-2*z^3*t^2-2*z^2*t-2*z^3+1-sqrt(1+4*z^3*t-4*z^2*t+z^2-2*z-4*z^3))/z^2/(z*t-t-z)^2: Gser:=simplify(series(G,z=0,18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser,z,n)) od: 1;1; for n from 2 to 15 do seq(coeff(P[n],t,j),j=0..floor(n/2)-1) od; # yields sequence in triangular form

G.f.: G = G(t,z) satisfies G = 1 + zG + z^2*(1 + zG + t(G-1-zG))^2 (see the Maple program for the explicit expression).

Showing 1-10 of 10 results.

cp's OEIS Frontend

A111808 Left half of trinomial triangle (A027907), triangle read by rows.

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A243827 Number A(n,k) of Dyck paths of semilength n having exactly one occurrence of the consecutive step pattern given by the binary expansion of k, where 1=U=(1,1) and 0=D=(1,-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A014531 Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 2nd column from the center.

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A025181 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 3. Also a(n) = T(n,n-3), where T is the array defined in A025177.

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A092107 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UUU's (triple rises) where U=(1,1). Rows have 1,1,1,2,3,4,5,... entries, respectively.

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A025568 a(n) = T(n,n+2) where T is the array defined in A025564.

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A026070 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 3. Also a(n) = T(n,n-3), where T is the array defined in A024996.

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A098470 Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 5th column from the center.

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A116401 Triangle whose k-th column has e.g.f. exp(x)*sum{j=0..k, Bessel_I(k+j,2x)}.

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