A199661 -id:A199661 - cp's OEIS frontend

A199597 Decimal expansion of x > 0 satisfying x^2 + x*cos(x) = sin(x).

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

Offset: 1

Clark Kimberling, Nov 08 2011

For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*cos(x)=c*sin(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 1.... 2.... A199597
... 1.... 3.... A199598
... 1.... 4.... A199599
... 2.... 1.... A199600
... 2.... 3.... A199601
... 2.... 4.... A199602
... 3.... 0.... A199603, A199604
... 3.... 1.... A199605, A199606
... 3.... 2.... A199607, A199608
... 3.... 3.... A199609, A199610
... 4.... 0.... A199611, A199612
... 4.... 1.... A199613, A199614
... 4.... 2.... A199615, A199616
... 4.... 3.... A199617, A199618
... 4.... 4.... A199619, A199620
... 1.... 0.... A199621
... 1.... 2.... A199622
... 1.... 3.... A199623
... 1.... 4.... A199624
... 2.... 1.... A199625
... 2.... 3.... A199661
... 1.... 0.... A199662
... 1.... 2.... A199663
... 1.... 3.... A199664
... 1.... 4.... A199665
... 2.... 0.... A199666
... 2.... 1.... A199667
... 2.... 3.... A199668
... 2.... 4.... A199669
.. -1.... 0.... A003957
.. -1.... 1.... A199722
.. -1.... 2.... A199721
.. -1.... 3.... A199720
.. -1.... 4.... A199719
.. -2.... 1.... A199726
.. -2.... 2.... A199725
.. -2.... 3.... A199724
.. -2.... 4.... A199723
.. -3.... 1.... A199730
.. -3.... 2.... A199729
.. -3.... 3.... A199728
.. -3.... 4.... A199727
.. -4.... 1.... A199737. A199738
.. -4.... 2.... A199735, A199736
.. -4.... 3.... A199733, A199734
.. -4.... 4.... A199731. A199732
.. -1.... 1.... A199742
.. -1.... 2.... A199741
.. -1.... 3.... A199740
.. -1.... 4.... A199739
.. -2.... 1.... A199776
.. -2.... 3.... A199775
.. -3.... 1.... A199780
.. -3.... 2.... A199779
.. -3.... 3.... A199778
.. -3.... 4.... A199777
.. -4.... 1.... A199782
.. -4.... 3.... A199781
.. -4.... 1.... A199786
.. -4.... 2.... A199785
.. -4.... 3.... A199784
.. -4.... 4.... A199783
.. -3.... 1.... A199789
.. -3.... 2.... A199788
.. -3.... 4.... A199787
.. -2.... 1.... A199793
.. -2.... 2.... A199792
.. -2.... 3.... A199791
.. -2.... 4.... A199790
.. -1.... 1.... A199797
.. -1.... 2.... A199796
.. -1.... 3.... A199795
.. -1.... 4.... A199794
.. -4.... 1.... A199873
.. -4.... 3.... A199872
.. -3.... 1.... A199871
.. -3.... 2.... A199870
.. -3.... 3.... A199869
.. -3.... 4.... A199868
.. -2.... 1.... A199867
.. -2.... 3.... A199866
.. -1.... 1.... A199865
.. -1.... 2.... A199864
.. -1.... 3.... A199863
.. -1.... 4.... A199862
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A199597, take f(x,u,v)=x^2+u*x*cos(x)-v*sin(x) and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

			1.1881851344514388032178109729076525973...

Index entries for transcendental numbers.

Cf. A199370, A199170, A198866, A198755, A198414, A197737, A199429.

(* Program 1:  A199597 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -Pi, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.18, 1.19}, WorkingPrecision -> 110]
RealDigits[r]  (* A199597 *)
(* Program 2: impl. surf. x^2+u*x*cos(x)=v*sin(x) *)
f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v*Sin[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .5, 3}]}, {u, 0, 2}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]]  (* for A199597 *)

Edited by Georg Fischer, Aug 03 2021

A112467 Riordan array ((1-2x)/(1-x), x/(1-x)).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, -1, -1, 1, 1, -1, -2, 0, 2, 1, -1, -3, -2, 2, 3, 1, -1, -4, -5, 0, 5, 4, 1, -1, -5, -9, -5, 5, 9, 5, 1, -1, -6, -14, -14, 0, 14, 14, 6, 1, -1, -7, -20, -28, -14, 14, 28, 20, 7, 1, -1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1, -1, -10, -44, -110, -165, -132, 0, 132, 165, 110

Offset: 0

Paul Barry, Sep 06 2005

Row sums are A000007. Diagonal sums are -F(n-2). Inverse is A112468. T(2n,n)=0.
(-1,1)-Pascal triangle. - Philippe Deléham, Aug 07 2006
Apart from initial term, same as A008482. - Philippe Deléham, Nov 07 2006
Each column equals the cumulative sum of the previous column. - Mats Granvik, Mar 15 2010
Reading along antidiagonals generates in essence rows of A192174. - Paul Curtz, Oct 02 2011
Triangle T(n,k), read by rows, given by (-1,2,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2011

			Triangle starts:
    1;
   -1,  1;
   -1,  0,   1;
   -1, -1,   1,   1;
   -1, -2,   0,   2,   1;
   -1, -3,  -2,   2,   3,   1;
   -1, -4,  -5,   0,   5,   4,  1;
   -1, -5,  -9,  -5,   5,   9,  5,  1;
   -1, -6, -14, -14,   0,  14, 14,  6,  1;
   -1, -7, -20, -28, -14,  14, 28, 20,  7,  1;
   -1, -8, -27, -48, -42,   0, 42, 48, 27,  8, 1;
   -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1;
  ...
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins:
   1,  1,
  -2, -1,  1,
   2,  0, -1,  1,
  -2,  0,  0, -1,  1,
   2,  0,  0,  0, -1,  1,
  -2,  0,  0,  0,  0, -1,  1,
   2,  0,  0,  0,  0,  0, -1,  1
  ... (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Restricting Dyck Paths and 312-avoiding Permutations, arXiv:2307.02837 [math.CO], 2023. Mentions this sequence.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
D. Foata and G.-N. Han, The doubloon polynomial triangle, Ram. J. 23 (2010), 107-126.
Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965. [Mentions application to design of antenna arrays. Annotated scan.]

Same first 3 rows as in A054525.

Cf. A008482, A037012, A080232, A112466, A112467, A174293, A174294, A174295, A174296, A174297.

[n eq 0 select 1 else (2*k-n)*Binomial(n,k)/n: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 04 2019

seq(seq( `if`(n=0, 1, (2*k-n)*binomial(n,k)/n), k=0..n), n=0..10); # G. C. Greubel, Dec 04 2019

T[n_, k_]= If[n==0, 1, ((2*k-n)/n)*Binomial[n, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019 *)

T(n, k) = if(n==0, 1, (2*k-n)*binomial(n,k)/n ); \\ G. C. Greubel, Dec 04 2019

def T(n, k):
    if (n==0): return 1
    else: return (2*k-n)*binomial(n,k)/n
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 04 2019

Number triangle T(n, k) = binomial(n, n-k) - 2*binomial(n-1, n-k-1).
Sum_{k=0..n} T(n, k)*x^k = (x-1)*(x+1)^(n-1). - Philippe Deléham, Oct 03 2005
T(n,k) = ((2*k-n)/n)*binomial(n, k), with T(0,0)=1. - Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019
T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(1,0)=-1, T(n,k)=0 for k>n or for n<0. - Philippe Deléham, Nov 01 2011
G.f.: (1-2x)/(1-(1+y)*x). - Philippe Deléham, Dec 15 2011
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A133494(n), A081294(n), A005053(n), A067411(n), A199661(n), A083233(n) for x = 1, 2, 3, 4, 5, 6, 7, respectively. - Philippe Deléham, Dec 15 2011
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 - x + x^2/2! + x^3/3!) = -1 - 2*x - 2*x^2/2! + 5*x^4/4! + 14*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014
Sum_{k=0..n} T(n,k) = 0^n = A000007(n). - G. C. Greubel, Dec 04 2019

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A199597 Decimal expansion of x > 0 satisfying x^2 + x*cos(x) = sin(x).

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A112467 Riordan array ((1-2x)/(1-x), x/(1-x)).

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