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Search: a108267 -id:a108267
Displaying 1-8 of 8 results found. page 1
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A108291 Triangle, read by rows, resulting from the matrix product of triangle A108267 with Pascal's triangle (A007318). +20
2
1, 2, 1, 9, 9, 1, 64, 96, 34, 1, 625, 1250, 750, 125, 1, 7776, 19440, 16470, 5265, 461, 1, 117649, 352947, 386561, 184877, 35329, 1715, 1, 2097152, 7340032, 9863168, 6307840, 1913408, 232288, 6434, 1, 43046721, 172186884, 274223556, 220016574 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Row sums form A108292. Column 0 is A000169(n) = (n+1)^n. Triangle with rows reversed is A108290.
LINKS
Table of n, a(n) for n=0..39.
EXAMPLE
Triangle begins:
1;
2,1;
9,9,1;
64,96,34,1;
625,1250,750,125,1;
7776,19440,16470,5265,461,1;
117649,352947,386561,184877,35329,1715,1;
2097152,7340032,9863168,6307840,1913408,232288,6434,1; ...
PROG
(PARI) {T(n, k)=local(X=x+x*O(x^(n-k))); polcoeff(sum(j=0, n, binomial(n+n*j+j, n*j+j)*(x/(1+X))^j)/(1+X), n-k)}
CROSSREFS
Cf. A108267, A060543, A108290, A000169.
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, May 31 2005
STATUS
approved
A108268 Column 2 of triangle A108267. +20
0
1, 31, 381, 3431, 26769, 193705, 1343521, 9091270, 60632419, 401001030, 2639871326, 17339260251, 113792427233, 746807661549, 4903854042841, 32227106641988, 211992209767971, 1395903036647155, 9200826759772935 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
LINKS
Table of n, a(n) for n=2..20.
PROG
(PARI)
a(n)=polcoeff((1-x)^(n+1)*sum(j=0, n, binomial(n+n*j+j, n*j+j)*x^j), 2)
CROSSREFS
Cf. A108267.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 29 2005
EXTENSIONS
Offset corrected by Andrey Zabolotskiy, Nov 23 2021
STATUS
approved
A060543 Triangle, read by antidiagonals, where T(n,k) = C(n+n*k+k, n*k+k). +10
11
1, 1, 1, 1, 3, 1, 1, 10, 5, 1, 1, 35, 28, 7, 1, 1, 126, 165, 55, 9, 1, 1, 462, 1001, 455, 91, 11, 1, 1, 1716, 6188, 3876, 969, 136, 13, 1, 1, 6435, 38760, 33649, 10626, 1771, 190, 15, 1, 1, 24310, 245157, 296010, 118755, 23751, 2925, 253, 17, 1, 1, 92378, 1562275 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Main diagonal is A108288. Antidiagonal sums is A108289. Inverse binomial transforms of each row give triangle A108290. G.f. of row n multiplied by (1-x)^(n+1) equals g.f. of row n of triangle A108267 (rows sums of A108267 equal (n+1)^n).
LINKS
Harry J. Smith, Table of n, a(n) for n=0..252
FORMULA
a(n) = A060539(n, k)/n = A007318(nk, k)/n = A060540(n, k)/A060540(n-1, k).
EXAMPLE
row 1: (2*n+1)/1!
row 2: (3*n+1)*(3*n+2)/2!
row 3: (4*n+1)*(4*n+2)*(4*n+3)/3!
row 4: (5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/4!
row 5: (6*n+1)*(6*n+2)*(6*n+3)*(6*n+4)*(6*n+5)/5!.
Table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
1,3,5,7,9,11,13,15,17,19,21,23,25,27,...
1,10,28,55,91,136,190,253,325,406,496,...
1,35,165,455,969,1771,2925,4495,6545,...
1,126,1001,3876,10626,23751,46376,82251,...
1,462,6188,33649,118755,324632,749398,...
1,1716,38760,296010,1344904,4496388,...
PROG
(PARI) T(n, k)=binomial(n+n*k+k, n*k+k)
(PARI) { i=0; write("b060543.txt", "0 1"); for (m=0, 20, for (k=0, m + 1, n=m - k + 1; write("b060543.txt", i++, " ", binomial(n + n*k + k, n*k + k))); ) } \\ Harry J. Smith, Jul 06 2009
CROSSREFS
Cf. A108290, A108267, A108288, A108289, A060544 (row 2), A015219 (row 3).
Rows include A000012, A001700, A025174. Columns include A000012, A005408, A060544. Main diagonal is A060545.
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Apr 02 2001
EXTENSIONS
Entry revised by Paul D. Hanna, May 31 2005
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007
STATUS
approved
A108290 Triangle, read by rows, such that row n equals the inverse binomial transform of row n of table A060543, where A060543(n,k) = C(n+n*k+k, n*k+k). +10
4
1, 1, 2, 1, 9, 9, 1, 34, 96, 64, 1, 125, 750, 1250, 625, 1, 461, 5265, 16470, 19440, 7776, 1, 1715, 35329, 184877, 386561, 352947, 117649, 1, 6434, 232288, 1913408, 6307840, 9863168, 7340032, 2097152, 1, 24309, 1513656, 18921924, 92365758, 220016574 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Row sums form A108292. Main diagonal is A000169(n) = (n+1)^n. Triangle with rows reversed is A108291.
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (1+n*x)*(2+n*x)*...*(n-1+n*x)/(n-1)! in the basis made of the binomial(x+i,i). - F. Chapoton, Nov 01 2022
LINKS
Table of n, a(n) for n=0..41.
EXAMPLE
BINOMIAL[1, 9, 9] = {1, 10, 28, 55, 91, 136, 190, 253, ...}.
BINOMIAL[1, 34, 96, 64] = {1, 35, 165, 455, 969, 1771, 2925, ...}.
BINOMIAL[1, 125, 750, 1250, 625] = {1, 126, 1001, 3876, 10626, ...}.
Triangle begins:
1;
1, 2;
1, 9, 9;
1, 34, 96, 64;
1, 125, 750, 1250, 625;
1, 461, 5265, 16470, 19440, 7776;
1, 1715, 35329, 184877, 386561, 352947, 117649;
1, 6434, 232288, 1913408, 6307840, 9863168, 7340032, 2097152; ...
PROG
(PARI) {T(n, k)=local(X=x+x*O(x^k)); polcoeff(sum(j=0, n, binomial(n+n*j+j, n*j+j)*(x/(1+X))^j)/(1+X), k)}
CROSSREFS
Cf. A108267, A060543, A108290, A000169.
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, May 31 2005
STATUS
approved
A108292 Row sums of triangle A108290. +10
3
1, 3, 19, 195, 2751, 49413, 1079079, 27760323, 822299383, 27565191753, 1031671508495, 42643092165765, 1929325374428791, 94835735736471369, 5032700868665421519, 286770182910733076163, 17463186681730290301671 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..16.
FORMULA
a(n) = Sum_{k=0..n} A108267(n, k)*2^k.
PROG
(PARI) a(n)=local(X=x+x*O(x^n)); sum(k=0, n, polcoeff(sum(j=0, n, binomial(n+n*j+j, n*j+j)*(x/(1+X))^j)/(1+X), k))
(PARI) a(n)=sum(k=0, n, 2^k*polcoeff( (1-x)^(n+1)*sum(j=0, n, binomial(n+n*j+j, n*j+j)*x^j), k))
CROSSREFS
Cf. A108267, A060543, A108290.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 31 2005
STATUS
approved
A108288 Main diagonal of table A060543; a(n) = C((n+1)^2-1, n*(n+1)). +10
2
1, 3, 28, 455, 10626, 324632, 12271512, 553270671, 28987537150, 1731030945644, 116068178638776, 8634941152058949, 705873715441872264, 62895036884524942320, 6067037854078498539696, 629921975126394617164575, 70043473196734767582082230 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..16.
FORMULA
a(n) = A060545(n+1). - R. J. Mathar, Aug 24 2008
PROG
(PARI) a(n)=binomial((n+1)^2-1, n*(n+1))
CROSSREFS
Cf. A060545, A108267, A060543, A108289.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 31 2005
STATUS
approved
A108289 Antidiagonal sums of table A060543. +10
2
1, 2, 5, 17, 72, 357, 2022, 12900, 91448, 711180, 6004981, 54619489, 531854438, 5515551251, 60642234815, 704106298738, 8603658260904, 110306422692488, 1479905106340895, 20727595895871297, 302423908621734606 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..20.
FORMULA
a(n)=Sum_{k=0..n} C(n+(n-k)*k, (n-k)*k+k)).
PROG
(PARI) a(n)=sum(k=0, n, binomial(n+(n-k)*k, (n-k)*k+k))
CROSSREFS
Cf. A108267, A060543, A108288.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 31 2005
STATUS
approved
A333829 Triangle read by rows: T(n,k) is the number of parking functions of length n with k strict descents. T(n,k) for n >= 1 and 0 <= k <= n-1. +10
1
1, 2, 1, 5, 10, 1, 14, 73, 37, 1, 42, 476, 651, 126, 1, 132, 2952, 8530, 4770, 422, 1, 429, 17886, 95943, 114612, 31851, 1422, 1, 1430, 107305, 987261, 2162033, 1317133, 202953, 4853, 1, 4862, 642070, 9613054, 35196634, 39471355, 13792438, 1262800, 16786, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
In a parking function w(1), ..., w(n), a strict descent is an index i such that w(i) > w(i+1).
Define an n-dimensional polytope as the convex hull of length n+1 nondecreasing parking functions. Then, the Ehrhart h*-polynomial of this polytope is Sum_{k=0..n-1} T(n,k) * z^(n-1-k).
LINKS
Table of n, a(n) for n=1..45.
Ari Cruz, Pamela E. Harris, Kimberly J. Harry, Jan Kretschmann, Matt McClinton, Alex Moon, John O. Museus, and Eric Redmon, On some discrete statistics of parking functions, arXiv:2312.16786 [math.CO], 2023.
Paul R. F. Schumacher, Descents in Parking Functions, J. Int. Seq. 21 (2018), #18.2.3.
EXAMPLE
The triangle T(n,k) begins:
n/k 0 1 2 3 4 5
1 1
2 2 1
3 5 10 1
4 14 73 37 1
5 42 476 651 126 1
6 132 2952 8530 4770 422 1
...
The 10 parking functions of length 3 with 1 strict descent are: [[1, 2, 1], [2, 1, 1], [1, 3, 1], [3, 1, 1], [2, 1, 2], [2, 2, 1], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2]].
PROG
(SageMath)
var('z, t')
assume(0<z<1)
# Returns a polynomial which is the generating function of strict descents in permutations of a multiset of integers. The multiplicity of these integers are given by an integer partition l. The function uses an analytic expression rather than enumerating the combinatorial objects.
def des_multiset(l):
return expand(factor(sum( mul( mul( t+i for i in range(1, k+1)) / factorial(k) for k in l ) * z**t , t , 0 , oo ) * (1-z)**(sum(l)+1) ))
# Returns the numbers of noncrossing partitions of size n and type l (an integer partition of n), cf. Kreweras: "Sur les partitions non-croisées d'un cycle".
def kreweras_numbers(l):
m = l.to_exp()
s = sum(l)
return ZZ.prod(range(s - len(l) + 2, s + 1)) // ZZ.prod(factorial(i) for i in m)
def Trow(n):
pol = sum(des_multiset(l) * kreweras_numbers(l) for l in Partitions(n))
return pol.list()
print([Trow(n) for n in (1..4)])
(SageMath) # using[latte_int from LattE]
# Install with "sage -i latte_int".
# Another method is to compute the Ehrhart h^*-polynomial of a polytope.
var('z, t')
def Tpol(n):
p = Polyhedron( NonDecreasingParkingFunctions(n+1) ).ehrhart_polynomial()
return expand(factor( (1-z)**(n+1) * sum( p * z**t , t , 0 , oo ) ))
def T(n, k):
return Tpol(n).list()[n-1-k]
CROSSREFS
Row sums give A000272(n+1).
Column k=0 gives A000108.
Similar to A108267.
KEYWORD
nonn,tabl,easy
AUTHOR
Matthieu Josuat-Vergès, Apr 06 2020
STATUS
approved
page 1

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Last modified August 29 06:09 EDT 2024. Contains 375510 sequences. (Running on oeis4.)