Yuriy Shablya - cp's OEIS frontend

User: Yuriy Shablya

Yuriy Shablya has authored 2 sequences.

A343960 Triangle read by rows: T(n,m) = Sum_{k=1..m} (k/n)binomial(n,m-k)binomial(n,m), n >= m >= 1.

1, 1, 2, 1, 5, 4, 1, 9, 17, 8, 1, 14, 46, 49, 16, 1, 20, 100, 180, 129, 32, 1, 27, 190, 510, 603, 321, 64, 1, 35, 329, 1225, 2121, 1827, 769, 128, 1, 44, 532, 2618, 6202, 7700, 5164, 1793, 256, 1, 54, 816, 5124, 15876, 26628, 25392, 13878, 4097, 512

Offset: 1

Yuriy Shablya, May 05 2021

			Triangle begins:
  ---------------------------------------------------------------------
   n \ m |     1     2     3     4     5     6     7     8     9    10
  -------+-------------------------------------------------------------
   1     |     1
   2     |     1     2
   3     |     1     5     4
   4     |     1     9    17     8
   5     |     1    14    46    49    16
   6     |     1    20   100   180   129    32
   7     |     1    27   190   510   603   321    64
   8     |     1    35   329  1225  2121  1827   769   128
   9     |     1    44   532  2618  6202  7700  5164  1793   256
   10    |     1    54   816  5124 15876 26628 25392 13878  4097   512

Cf. A001263.

T[n_, m_] := Sum[Binomial[n, m - k] * Binomial[n, m] * k/n, {k, 1, n}]; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

T(n,m):=sum((k/n)*binomial(n,m-k)*binomial(n,m),k,1,m)

T(n,m) = Sum_{k=1..m} (k/n)*binomial(n,m-k)*binomial(n,m).
G.f.: N(x,y)/(1-N(x,y)), where N(x,y) is a g.f. for the Narayana numbers A001263.
T(n, m) = A001263(n, m)*hypergeom([1 - m, 2], [n - m + 2], -1). - Peter Luschny, May 06 2021

A316773 Triangle read by rows: T(n,m) = Sum_{k=m+1..n} (n-1)!/(k-1)!binomial(2n-k-1, n-1)*E(k,m) where E(n,m) is Euler's triangle A173018, T(0,0) = 1, n >= m >= 0.

Original entry on oeis.org

1, 1, 0, 3, 1, 0, 19, 10, 1, 0, 193, 119, 23, 1, 0, 2721, 1806, 466, 46, 1, 0, 49171, 34017, 10262, 1502, 87, 1, 0, 1084483, 770274, 255795, 47020, 4425, 162, 1, 0, 28245729, 20429551, 7235853, 1539939, 193699, 12525, 303, 1, 0, 848456353, 621858526, 230629024, 54314242, 8273758, 755170, 34912, 574, 1, 0

Offset: 0

Yuriy Shablya, Sep 13 2018

T(n,m) is the number of labeled binary trees of size n with m ascents on the left branch.

			Triangle begins:
--------------------------------------------------------------------------
n\k|       0         1         2        3       4      5     6   7   8   9
------+-------------------------------------------------------------------
0 |         1
1 |         1         0
2 |         3         1         0
3 |        19        10         1        0
4 |       193       119        23        1       0
5 |      2721      1806       466       46       1      0
6 |     49171     34017     10262     1502      87      1     0
7 |   1084483    770274    255795    47020    4425    162     1   0
8 |  28245729  20429551   7235853  1539939  193699  12525   303   1  0
9 | 848456353 621858526 230629024 54314242 8273758 755170 34912 574  1  0

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Yuriy Shablya, Dmitry Kruchinin, Vladimir Kruchinin, Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application, Mathematics (2020) Vol. 8, No. 6, 962.

Cf. A033184, A008279, A173018.

T := (n,m) -> `if`(n=0, 1, add((n-1)!/(k-1)!*binomial(2*n-k-1, n-1)*
combinat:-eulerian1(k, m), k = m+1..n)):
for n from 0 to 6 do seq(T(n, k), k=0..n) od; # Peter Luschny, Sep 04 2020

Table[Boole[n == 0] + Sum[(n - 1)!/(k - 1)!*Binomial[2 n - k - 1, n - 1]*Sum[(-1)^j*(m + 1 - j)^k*Binomial[k + 1, j], {j, 0, m}], {k, m + 1, n}], {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Sep 04 2020 *)

T(n,m):=if m>n then 0 else if n=0 then 1 else sum((n-1)!/(k-1)!*binomial(2*n-k-1,n-1)*sum((-1)^j*(m+1-j)^k*binomial(k+1,j),j,0,m),k,m+1,n);

E.g.f.: Sum_{n >= m >= 0} T(n, m)/n! * x^n * y^m = E(C(x),y) = (y-1)/(y-exp(C(x)*(y-1))), where E(x,y) is an e.g.f. for Euler's triangle A173018.

T(n,m) = Sum_{k = m+1..n} C(n,k)*E(k,m)*P(n,n-k), T(0,0)=1, where C(n,m) is the transposed Catalan's triangle A033184, E(n,m) is Euler's triangle A173018, and P(n,m) is the number of k-permutations of n A008279.

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User: Yuriy Shablya

A343960 Triangle read by rows: T(n,m) = Sum_{k=1..m} (k/n)binomial(n,m-k)binomial(n,m), n >= m >= 1.

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A316773 Triangle read by rows: T(n,m) = Sum_{k=m+1..n} (n-1)!/(k-1)!binomial(2n-k-1, n-1)*E(k,m) where E(n,m) is Euler's triangle A173018, T(0,0) = 1, n >= m >= 0.

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cp's OEIS Frontend

User: Yuriy Shablya

A343960 Triangle read by rows: T(n,m) = Sum_{k=1..m} (k/n)*binomial(n,m-k)*binomial(n,m), n >= m >= 1.

Views

Author

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Examples

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Maxima

Formula

A316773 Triangle read by rows: T(n,m) = Sum_{k=m+1..n} (n-1)!/(k-1)!*binomial(2*n-k-1, n-1)*E(k,m) where E(n,m) is Euler's triangle A173018, T(0,0) = 1, n >= m >= 0.

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A343960 Triangle read by rows: T(n,m) = Sum_{k=1..m} (k/n)binomial(n,m-k)binomial(n,m), n >= m >= 1.

A316773 Triangle read by rows: T(n,m) = Sum_{k=m+1..n} (n-1)!/(k-1)!binomial(2n-k-1, n-1)*E(k,m) where E(n,m) is Euler's triangle A173018, T(0,0) = 1, n >= m >= 0.