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A127717 Triangle read by rows. T(n, k) = k * binomial(n + 1, k + 1), for 1 <= k <= n. 2
1, 3, 2, 6, 8, 3, 10, 20, 15, 4, 15, 40, 45, 24, 5, 21, 70, 105, 84, 35, 6, 28, 112, 210, 224, 140, 48, 7, 36, 168, 378, 504, 420, 216, 63, 8, 45, 240, 630, 1008, 1050, 720, 315, 80, 9, 55, 330, 990, 1848, 2310, 1980, 1155, 440, 99, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
T(n,k) is the sum of the greatest element in each size k subset of {1,2,...,n}. - Geoffrey Critzer, Oct 17 2009
Reversed unsigned rows of A055137 with the diagonal and first subdiagonal removed. - Tom Copeland, Nov 04 2012
Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x-1)^0 + A_1*(x-1)^1 + A_2*(x-1)^2 + ... + A_n*(x-1)^n. This sequence gives A_0, ..., A_n as the entries in the n-th row of this triangle, starting at n = 0. - Derek Orr, Oct 30 2014
LINKS
Cyann Donnot, Antoine Genitrini, Yassine Herida, Unranking Combinations Lexicographically: an efficient new strategy compared with others, hal-02462764 [cs] / [cs.DS] / [math] / [math.CO], 2020.
Antoine Genitrini and Martin Pépin, Lexicographic unranking of combinations revisited, hal-03040740v2 [cs.DM] [cs.DS] [math.CO], 2020.
FORMULA
A002260 * A007318 (Pascal's Triangle), where A002260 = the matrix [1; 1,2; 1,2,3,...].
T(n,k) = Sum_{i=k..n} binomial(i-1, k-1)*i. - Geoffrey Critzer, Oct 17 2009
Row sums = A000337: (1, 5, 17, 49, 129, ...) A007318 * A002260 = A127718.
From Geoffrey Critzer, Oct 18 2009: (Start)
T(n,k) = k*binomial(n+1, k+1).
Recurrence for column k: a(n) = a(n-1) + n*binomial(n-1, k-1) = a(n-1) + k*binomial(n, k).
O.g.f. for column k: k*x^k/(1-x)^(k+2). (End)
T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n+2-k, k+2-i). - Mircea Merca, Apr 11 2012
G.f.: 1/((1 - x)*(1 - x - x*y)^2), assuming the triangle (0,0)-based. - Vladimir Kruchinin, Jan 07 2023
EXAMPLE
First few rows of the triangle:
[1 2 3 4 5 6 7 8 9]
[1] 1;
[2] 3, 2;
[3] 6, 8, 3;
[4] 10, 20, 15, 4;
[5] 15, 40, 45, 24, 5;
[6] 21, 70, 105, 84, 35, 6;
[7] 28, 112, 210, 224, 140, 48, 7;
[8] 36, 168, 378, 504, 420, 216, 63, 8;
[9] 45, 240, 630, 1008, 1050, 720, 315, 80, 9;
...
T(4, 3) = 15 because the size 3 subsets of {1, 2, 3, 4} are {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}. Adding the largest element from each subset we get 3 + 4 + 4 + 4 = 15. - Geoffrey Critzer, Oct 17 2009
MAPLE
# Assuming (1, 1)-based triangle:
T := (n, k) -> k*binomial(n+1, k+1):
seq(seq(T(n, k), k = 1..n), n = 1..9);
# Assuming (0, 0)-based triangle:
gf := 1/((1 - x)*(1 - x - x*y)^2): ser := series(gf, x, 11):
seq(seq(coeff(coeff(ser, x, n), y, k), k=0..n), n=0..9); # Peter Luschny, Jan 07 2023
MATHEMATICA
Table[Table[Sum[Binomial[i - 1, k - 1]*i, {i, k, n}], {k, 1, n}], {n, 1, 10}] // Grid (* Geoffrey Critzer, Oct 17 2009 *)
PROG
(PARI) T(n, k) = k*sum(i=0, n-k, binomial(i+k, k))
for(n=1, 15, for(k=1, n, print1(T(n, k), ", "))) \\ Derek Orr, Oct 30 2014
CROSSREFS
Sequence in context: A293204 A273344 A370665 * A210236 A193998 A209171
KEYWORD
nonn,tabl,easy
AUTHOR
Gary W. Adamson, Jan 25 2007
EXTENSIONS
a(8) = 20, corrected by Geoffrey Critzer, Oct 17 2009
More terms from Derek Orr, Oct 30 2014
Offset set to 1 and new name using a formula of Geoffrey Critzer by Peter Luschny, Jan 07 2023
STATUS
approved

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Last modified June 21 03:10 EDT 2024. Contains 373535 sequences. (Running on oeis4.)