A257241 -id:A257241 - cp's OEIS frontend

A258143 Row sums of A257241, Stifel's version of the arithmetical triangle.

1, 2, 6, 10, 25, 41, 98, 162, 381, 637, 1485, 2509, 5811, 9907, 22818, 39202, 89845, 155381, 354521, 616665, 1401291, 2449867, 5546381, 9740685, 21977515, 38754731, 87167163, 154276027, 345994215, 614429671, 1374282018, 2448023842, 5461770405, 9756737701, 21717436833

Offset: 1

Wolfdieter Lang, May 22 2015

a(n) is the number of nonempty subsets of {1,2,...,n} that contain either more odd than even numbers or the same number of odd and even numbers. For example, for n=4, a(4)=10 and the 10 subsets are {1}, {3}, {1,3}, {1,2,3}, {1,3,4}; {1,2}, {1,4}, {2,3}, {3,4}, {1,2,3,4}. - Enrique Navarrete, Dec 16 2019

			n=3: a(3) = 2^3 - (1 + A008549(1)) = 8 - (1 + 1) = 6.
n=4: a(4) = 2^4 - (1 + A000346(1)) = 16 - (1 +  5) = 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A257241, A000346, A008549, A000108, A116406, A258144.

a258143 = sum . a257241_row  -- Reinhard Zumkeller, May 22 2015

Table[Sum[Binomial[n, m], {m, Ceiling[n/2]}], {n, 50}] (* Paolo Xausa, Nov 14 2024 *)

a(n) = Sum_{m = 1 .. ceiling(n/2)} binomial(n, m), n >= 1.
a(n) = 2^n - 2 - Sum_{i=1..floor(n/2)-1} binomial(n, i), n >= 2; a(1)=1. - Enrique Navarrete, Dec 16 2019
a(2*k+1) = 2^(2*k+1) - (1 + A008549(k)), k >= 0.
a(2*k) = 2^(2*k) - (1 + A000346(k-1)), k >= 1.
O.g.f.: x*(2+3*x+x^2 - (1-x^2)*(1+x)*c(x^2))/((1-(2*x)^2)*(1-x^2)) where c(x) is the o.g.f. of A000108.
O.g.f. for a(2*k+1), k >= 0: (2+x - (1-x)*c(x))/ ((1-4*x)*(1-x)).
O.g.f. for a(2*(k+1)), k >= 0: (3 - (1-x)*c(x))/ ((1-4*x)*(1-x)).
a(n) = A116406(n+1) - 1. - Hugo Pfoertner, Nov 14 2024

A258144 Alternating row sums of A257241, Stifel's version of the arithmetical triangle.

Original entry on oeis.org

1, 2, 0, -2, 5, 11, -14, -34, 57, 127, -209, -461, 793, 1717, -3002, -6434, 11441, 24311, -43757, -92377, 167961, 352717, -646645, -1352077, 2496145, 5200301, -9657699, -20058299, 37442161, 77558761, -145422674, -300540194, 565722721, 1166803111, -2203961429

Offset: 1

Wolfdieter Lang, May 22 2015

			n = 3: a(3) = (1 - A001791(1)) = 1 - 1 = 0.
n = 4: a(4) = (1 - A001700(1)) = 1 - 3 = -2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A257241, A001700, A001791, A258143, A000108.

Cf. A033999.

a258144 = sum . zipWith (*) (cycle [1, -1]) . a257241_row
-- Reinhard Zumkeller, May 22 2015

Table[Sum[(-1)^(m+1)*Binomial[n, m], {m, Ceiling[n/2]}], {n, 50}] (* Paolo Xausa, Nov 14 2024 *)

a(n) = Sum_{m = 1 .. ceiling(n/2)} (-1)^(m+1)* binomial(n, m), n >= 1.
a(2*k+1) = (1 - (-1)^(k+1)*A001791(k)), k >= 0.
a(2*k) =  (1 - (-1)^k*A001700(k-1)), k >= 1.
O.g.f. for a(2*k+1), k >= 0: (2+3*x - (1-x)*(1+2*x)*c(-x))/((1+4*x)*(1-x)), with the o.g.f. c(x) of A000108 (Catalan).
O.g.f. for a(2*(k+1)), k >= 0:
  (3+2*x - (1-x)*c(-x))/((1+4*x)*(1-x)).
O.g.f. for a(n), n >= 1:
x*((1+x)*(2+x+2*x^2) - (1+x+2*x^2)*(1-x^2)*c(-x^2))/((1+4*x^2)*(1-x^2)).

A014410 Elements in Pascal's triangle (by row) that are not 1.

Original entry on oeis.org

2, 3, 3, 4, 6, 4, 5, 10, 10, 5, 6, 15, 20, 15, 6, 7, 21, 35, 35, 21, 7, 8, 28, 56, 70, 56, 28, 8, 9, 36, 84, 126, 126, 84, 36, 9, 10, 45, 120, 210, 252, 210, 120, 45, 10, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 13, 78

Offset: 2

Mohammad K. Azarian

Also, rows of triangle formed using Pascal's rule except begin and end n-th row with n+2. - Asher Auel.
Row sums are A000918. - Roger L. Bagula and Gary W. Adamson, Jan 15 2009
Given the triangle signed by rows (+ - + ...) = M, with V = a variant of the Bernoulli numbers starting [1/2, 1/6, 0, -1/30, 0, 1/42, ...]; M*V = [1, 1, 1, ...]. - Gary W. Adamson, Mar 05 2012
Also A014410 * [1/2, 1/6, 0, -1/30, 0, 1/42, 0, ...] = [1, 2, 3, 4, ...]. For an alternative way to derive the Bernoulli numbers from a modified version of Pascal's triangle see A135225. - Peter Bala, Dec 18 2014
T(n,k) mod n = A053201(n,k), k=1..n-1. - Reinhard Zumkeller, Aug 17 2013
From Wolfdieter Lang, May 22 2015: (Start)
This is Johannes Scheubel's (1494-1570) (also Scheybl, Schöblin) version of the arithmetical triangle from his 1545 book "De numeris et diversis rationibus". See the Kac reference, p. 396 and the Table 12.1 on p. 395.
The row sums give 2*A000225(n-1) = A000918(n) = 2*(2^n - 1), n >= 2. (See the second comment above).
The alternating row sums give repeat(2,0) = 2*A059841(n), n >= 2. (End)
T(n+1,k) is the number of k-facets of the n-simplex. - Jianing Song, Oct 22 2023

			The triangle T(n,k) begins:
n\k  1  2   3   4    5    6    7    8   9  10 11
2:   2
3:   3  3
4:   4  6   4
5:   5 10  10   5
6:   6 15  20  15    6
7:   7 21  35  35   21    7
8:   8 28  56  70   56   28    8
9:   9 36  84 126  126   84   36    9
10: 10 45 120 210  252  210  120   45  10
11: 11 55 165 330  462  462  330  165  55  11
12: 12 66 220 495  792  924  792  495 220  66 12
... reformatted. - _Wolfdieter Lang_, May 22 2015

Victor J. Kac, A History of Mathematics, third edition, Addison-Wesley, 2009, pp. 395, 396.

Reinhard Zumkeller, Rows n=2..150 of triangle, flattened
Carl McTague, On the Greatest Common Divisor of binomial(qn, q), binomial(qn,2q), ..., binomial(qn, qn-q), arXiv:1510.06696 [math.CO], 2015.
Wikipedia, Johannes Scheubel (in German).
Wikipedia, Simplex

Cf. A007318, A000918, A027641.
A180986 is the same sequence but regarded as a square array.
Cf. A000225,A059841, A257241 (Stifel's version).

a014410 n k = a014410_tabl !! (n-2) !! (k-1)
a014410_row n = a014410_tabl !! (n-2)
a014410_tabl = map (init . tail) $ drop 2 a007318_tabl
-- Reinhard Zumkeller, Mar 12 2012

for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i-1) od; # Zerinvary Lajos, Dec 02 2007

Select[ Flatten[ Table[ Binomial[ n, i ], {n, 0, 13}, {i, 0, n} ] ], #>1& ]

T(n,k) = binomial(n,k) = A007318(n,k), n >= 2, k = 1, 2, ..., n-1.
a(n) = C(A003057(n),A002260(n)) = C(A003057(n),A004736(n)). - Lekraj Beedassy, Jul 29 2006
T(n,k) = A028263(n,k) - A007318(n,k). - Reinhard Zumkeller, Mar 12 2012
gcd_{k=1..n-1} T(n, k) = A014963(n), see Theorem 1 of McTague link. - Michel Marcus, Oct 23 2015

More terms from Erich Friedman

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A258143 Row sums of A257241, Stifel's version of the arithmetical triangle.

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A258144 Alternating row sums of A257241, Stifel's version of the arithmetical triangle.

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A014410 Elements in Pascal's triangle (by row) that are not 1.

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