A007019 -id:A007019 - cp's OEIS frontend

A377825 Number of distinct permutations of the terms of the n-th row of Pascal's triangle with alternating signs.

1, 2, 3, 24, 30, 720, 630, 40320, 22680, 3628800, 1247400, 479001600, 97297200, 87178291200, 10216206000, 20922789888000, 1389404016000, 6402373705728000, 237588086736000, 2432902008176640000, 49893498214560000, 1124000727777607680000, 12623055048283680000

Offset: 0

Ryan Jean, Nov 08 2024

Note that for any given n, there are n+1 terms in that row.

			For n = 0, a(0) = 1 since there is just one term.
For n = 1, the signed row terms are {1, -1} so a(1) = 2 permutations.
For n = 2, the signed row terms are {1, -2, 1} which have only a(2) = 3 distinct permutations.
For n = 3, the signed row terms are {1, -3, 3, -1} which have a(3) = 24 permutations.

Paolo Xausa, Table of n, a(n) for n = 0..400

Bisections are: A007019, A010050.

Cf. A007318, A072345, A142150.

seq((n+1)! / (2^((n*(1+(-1)^n)) / 4)), n=0..22); # Georg Fischer, Dec 19 2024

A377825[n_] := (n+1)!/2^((n*(1 + (-1)^n))/4); Array[A377825, 25, 0] (* Paolo Xausa, Dec 20 2024 *)

a(n) = (n+1)! / (2^((n*(1+(-1)^n)) / 4)).
E.g.f.: 2*(x^6+x^5-4*x^3-3*x^2+4*x+2)/((x-1)^2*(x+1)^2*(x^2-2)^2). - Alois P. Heinz, Nov 09 2024
a(n) = (n+1)!/A072345(n-1) for n > 0. - Stefano Spezia, Nov 09 2024
Sum_{n>=0} 1/a(n) = cosh(1) + sinh(sqrt(2))/sqrt(2) - 1. - Amiram Eldar, Dec 25 2024

a(22) corrected by Georg Fischer, Dec 19 2024

A320842 Regular triangle whose rows are the coefficients of the Dominici expansion of f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t.

Original entry on oeis.org

1, 7, 3, 127, 123, 30, 4369, 6822, 3579, 630, 243649, 532542, 439899, 162630, 22680, 20036983, 56717781, 64697499, 37155267, 10735470, 1247400, 2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200, 343141433761, 1427877062076, 2563294235106, 2572662311496, 1558544277681, 569674791180, 116270210700, 10216206000

Offset: 1

Matthew Miller, Dec 11 2018

It appears that the first column (7, 127, 4369, ...) is from the sequence A002067.
It appears that the diagonal (3, 30, 630, ...) is from the sequence A007019.
It appears as though the unsigned row sum (10, 280, 15400, ...) is from the sequence A025035.
It appears as though the alternating sign row sum (sum(7, -3) = 4, sum(-127, 123, -30) = -34, ...) is from the sequence A002105.
This triangular array arises as the coefficients from terms in the inverse expansion of the function f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t evaluated at t = 0 for even values of the operation, using a method of Dominici's (nested derivatives, referenced below).
Without proof, appears to be related to computing the 'critical t-value' of Student's t-distribution. (conj.) Critical t-value t_(v, beta) is equal to: sqrt((v/(1-S^2)) - v) where S = (1/2)*Sum_{k>=1} (D^(2*k-2)[f](0)*(1/(2*k-1)!)*(B(1/2, v/2)*(1-2*beta))^(2*k-1)); where (1 - beta) is the confidence interval 'atta' (for a one-tailed distribution such that 'cumulative probability' = t_atta, where beta = 1-atta), x = 1 - (v/2), v: degrees of freedom, B(1/2, v/2) = gamma(1/2)*gamma(v/2)/gamma(1/2 + v/2), D^(2*k - 2)[f](0) is a polynomial function of 'x' whose coefficients are the terms of this sequence as computed using a method of Dominici's on f(t,x) with respect to t (referenced below).

			Given D^k[f]_(b) = (d/dt [f(t)*D^(k-1)[f](t)])_t = b where D^0[f](b) = 1, then for f(t,x) = (1/2)*(1 - t^2)^(-x) where f(0) = 1/2 one obtains: D^2[f]_(0) = -x/2, D^4[f]_(0) = (x/4)*(7*x - 3), D^6[f]_(0) = -(x/8)*(127*x^2 - 123*x + 30), etc., where b is an arbitrary constant.
Triangle begins:
           1;
           7,          3;
         127,        123,          30;
        4369,       6822,        3579,        630;
      243649,     532542,      439899,     162630,      22680;
    20036983,   56717781,    64697499,   37155267,   10735470,   1247400;
  2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200;
         ...

Diego Dominici, Nested derivatives: a simple method for computing series expansions of inverse functions, International Journal of Mathematics and Mathematical Sciences, Volume 2003, Issue 58, Pages 3699-3715.
Wikipedia, Student's t-distribution

Cf. A002067, A007019, A025035, A002105.

A347201 Decimal expansion of sin(sqrt(2)).

Original entry on oeis.org

9, 8, 7, 7, 6, 5, 9, 4, 5, 9, 9, 2, 7, 3, 5, 5, 2, 7, 0, 6, 9, 1, 3, 4, 0, 7, 2, 0, 7, 8, 9, 4, 2, 6, 5, 5, 9, 0, 6, 7, 9, 3, 1, 2, 9, 5, 1, 6, 0, 3, 7, 1, 9, 6, 0, 3, 8, 7, 6, 3, 6, 8, 5, 5, 0, 2, 9, 1, 3, 7, 2, 3, 0, 9, 3, 0, 1, 1, 6, 2, 9, 3, 9, 9, 7, 6, 2

Offset: 0

Sean A. Irvine, Aug 22 2021

			0.98776594599273552706913407207894265590...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002193, A007019, A049469.

RealDigits[Sin[Sqrt[2]], 10, 120][[1]] (* Amiram Eldar, Jun 01 2023 *)

Equals sqrt(2) * Sum_{n>=0} (-1)^n/A007019(n) = sqrt(2) * Sum_{n>=0} (-2)^n/(2*n+1)!. - Amiram Eldar, Jun 01 2023

cp's OEIS Frontend

A377825 Number of distinct permutations of the terms of the n-th row of Pascal's triangle with alternating signs.

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A320842 Regular triangle whose rows are the coefficients of the Dominici expansion of f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t.

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Author

Keywords

Comments

Examples

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Crossrefs

A347201 Decimal expansion of sin(sqrt(2)).

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula