Real-rooted atlas

A232535 → basis A007318 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(12 + 6\,x + 3\,x^{2} - 60\,n - 2\,n x + n x^{2} + 96\,n^{2} - 88\,n^{2} x - 19\,n^{2} x^{2} - 48\,n^{3} + 172\,n^{3} x + 9\,n^{3} x^{2} + n^{3} x^{3} - 72\,n^{4} x + 40\,n^{4} x^{2} + 4\,n^{4} x^{3} - 12\,n^{5} x^{2} + 4\,n^{5} x^{3}\right)\,P_{n}(x) = \left(48 + 152\,x + 152\,x^{2} + 38\,x^{3} - x^{4} - 240\,n - 664\,n x - 556\,n x^{2} - 122\,n x^{3} - 5\,n x^{4} + 384\,n^{2} + 736\,n^{2} x + 322\,n^{2} x^{2} - 5\,n^{2} x^{4} - 192\,n^{3} + 112\,n^{3} x + 524\,n^{3} x^{2} + 223\,n^{3} x^{3} + 31\,n^{3} x^{4} - 288\,n^{4} x - 200\,n^{4} x^{2} + 84\,n^{4} x^{3} + 40\,n^{4} x^{4} - 48\,n^{5} x^{2} - 20\,n^{5} x^{3} + 12\,n^{5} x^{4}\right)\,P_{n-1}(x) + \left(24\,x + 76\,x^{2} + 70\,x^{3} + 16\,x^{4} - 2\,x^{5} - 96\,n x - 256\,n x^{2} - 226\,n x^{3} - 80\,n x^{4} - 14\,n x^{5} + 96\,n^{2} x + 112\,n^{2} x^{2} - 62\,n^{2} x^{3} - 116\,n^{2} x^{4} - 38\,n^{2} x^{5} + 144\,n^{3} x^{2} + 160\,n^{3} x^{3} - 16\,n^{3} x^{4} - 32\,n^{3} x^{5} + 24\,n^{4} x^{3} + 16\,n^{4} x^{4} - 8\,n^{4} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
212562795
34832033364765
41921664266412161255861
576881921771213056337521643319
6307238912105984108544469007848343311603
712288180224592128778240477600137088161215122194201
84915281920031518725062656404800016796163457443020872915187177
91966083670016161832963073638430368000166256645019168780288527311000103633155
10786432162529288080588817720934420876800014257152057401344132464641614492870001331698744743
113145728713031683945922569814671361343897600110238105655819545617337548831691088311520013709317284663302997
1212582912310378496189215539252638515208218214400788122828848245227521898184704470733120700128005675384207936219730847749901
13503316481342177280893701324827497857024482263040005298506956838118506496182357852165764981248117145600014479150498542083053832744202484435567
14201326592577136230441674604544140525961216273580032000338988367872280603459584158506942464612440248321595443200027108956162832261121642477243629633751320061499579
Technical details
Node id
n-72def76e240cfea1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-62f5da75567e6122

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      3,       8;
  2:     12,      56,       27;
  3:     48,     320,      333,       64;
  4:    192,    1664,     2664,     1216,      125;
  5:    768,    8192,    17712,    13056,     3375,      216;
  6:   3072,   38912,   105984,   108544,    46900,     7848,     343;
  7:  12288,  180224,   592128,   778240,   477600,   137088,   16121,    512;
  8:  49152,  819200,  3151872,  5062656,  4048000,  1679616,  345744,  30208,   729;
  9: 196608, 3670016, 16183296, 30736384, 30368000, 16625664, 5019168, 780288, 52731, 1000;
%F Formula
The row polynomials satisfy -(48*n^3-96*n^2+60*n-12+(72*n^4-172*n^3+88*n^2+2*n-6)*x+(12*n^5-40*n^4-9*n^3+19*n^2-n-3)*x^2-(4*n^5+4*n^4+n^3)*x^3)*P(n,x) = -(192*n^3-384*n^2+240*n-48+(288*n^4-112*n^3-736*n^2+664*n-152)*x+(48*n^5+200*n^4-524*n^3-322*n^2+556*n-152)*x^2+(20*n^5-84*n^4-223*n^3+122*n-38)*x^3-(12*n^5+40*n^4+31*n^3-5*n^2-5*n-1)*x^4)*P(n-1,x) + ((96*n^2-96*n+24)*x+(144*n^3+112*n^2-256*n+76)*x^2+(24*n^4+160*n^3-62*n^2-226*n+70)*x^3+(16*n^4-16*n^3-116*n^2-80*n+16)*x^4-(8*n^4+32*n^3+38*n^2+14*n+2)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(192*n^3 - 384*n^2 + 240*n - 48 + (288*n^4 - 112*n^3 - 736*n^2 + 664*n - 152)*x + (48*n^5 + 200*n^4 - 524*n^3 - 322*n^2 + 556*n - 152)*x^2 + (20*n^5 - 84*n^4 - 223*n^3 + 122*n - 38)*x^3 - (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^4)*P[n - 1, x] + ((96*n^2 - 96*n + 24)*x + (144*n^3 + 112*n^2 - 256*n + 76)*x^2 + (24*n^4 + 160*n^3 - 62*n^2 - 226*n + 70)*x^3 + (16*n^4 - 16*n^3 - 116*n^2 - 80*n + 16)*x^4 - (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^5)*D[P[n - 1, x], x])/(-48*n^3 + 96*n^2 - 60*n + 12 - (72*n^4 - 172*n^3 + 88*n^2 + 2*n - 6)*x - (12*n^5 - 40*n^4 - 9*n^3 + 19*n^2 - n - 3)*x^2 + (4*n^5 + 4*n^4 + n^3)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(192*n^3 - 384*n^2 + 240*n - 48 + (288*n^4 - 112*n^3 - 736*n^2 + 664*n - 152)*x + (48*n^5 + 200*n^4 - 524*n^3 - 322*n^2 + 556*n - 152)*x^2 + (20*n^5 - 84*n^4 - 223*n^3 + 122*n - 38)*x^3 - (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^4)*v[i - 1] + ((96*n^2 - 96*n + 24)*x + (144*n^3 + 112*n^2 - 256*n + 76)*x^2 + (24*n^4 + 160*n^3 - 62*n^2 - 226*n + 70)*x^3 + (16*n^4 - 16*n^3 - 116*n^2 - 80*n + 16)*x^4 - (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^5)*deriv(v[i - 1], x))/(-48*n^3 + 96*n^2 - 60*n + 12 - (72*n^4 - 172*n^3 + 88*n^2 + 2*n - 6)*x - (12*n^5 - 40*n^4 - 9*n^3 + 19*n^2 - n - 3)*x^2 + (4*n^5 + 4*n^4 + n^3)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023