Real-rooted atlas

A207606 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-18 + 54\,x - 12\,n + 90\,n x + 6\,n^{2} + 36\,n^{2} x\right)\,P_{n}(x) = \left(-99 + 189\,x + 324\,x^{2} - 6\,n + 252\,n x + 270\,n x^{2} + 9\,n^{2} + 63\,n^{2} x + 54\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-5 + 42\,x + 207\,x^{2} + 108\,x^{3} + 5\,n + 48\,n x + 117\,n x^{2} + 54\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 33 + 36\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0333669
1132306162600
2465167418636484650
315307506129609396243033822
44824300247079479380425258748236295
514772111510334530511920422820179334306181605504
644283393138143459127945003145635205140671952310497610688052
7130620133360257383641360994419493460172991709307872278186435429470049190
8380415439138821781710609887881063720351200225608775118840153320104516731180980453474057
9109659614123052793680122564832605283306007257078366740246524185393121665181803838185038972342906470584
103134448445573442798760601025933040244043199039590911084474300194352640628019016730456690251701383518071275458418475535070
118896002138355668960768216394082010010640214270199371206522660569480825501137336174401221508307603810261894124849105148441452398116632178142
122509624542388567232250290941463960012444258286195941570422201451227065121641965307841364850319148238752676035133564510100314803161720274517133923132731939977995
1370431540128387881810621804590528780966481770331447804218638363107383120724989646520202009461782649886938050828503746605905901445859642243772168218059595793744304672104570056916800
14196772109385034002234417839357186481908180685134010575180875378791835463369380675261115206952595775033122551529552717503378404980665164988834519658134915575113961486511020447192475137749507228408074440424
Technical details
Node id
n-93da713ca5d063cd · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c8dfaec64c280b41

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:      33,       36;
  1:     132,      306,      162;
  2:     465,     1674,     1863,       648;
  3:    1530,     7506,    12960,      9396,      2430;
  4:    4824,    30024,    70794,     79380,     42525,      8748;
  5:   14772,   111510,   334530,    511920,    422820,    179334,     30618;
  6:   44283,   393138,  1434591,   2794500,   3145635,   2051406,    719523,    104976;
  7:  130620,  1333602,  5738364,  13609944,  19493460,  17299170,   9307872,   2781864,    354294;
  8:  380415,  4391388, 21781710,  60988788, 106372035, 120022560,  87751188,  40153320,  10451673,  1180980;
  9: 1096596, 14123052, 79368012, 256483260, 528330600, 725707836, 674024652, 418539312, 166518180, 38381850, 3897234;
%F Formula
The row polynomials satisfy (6*n^2-12*n-18+(36*n^2+90*n+54)*x)*P(n,x) = (9*n^2-6*n-99+(63*n^2+252*n+189)*x+(54*n^2+270*n+324)*x^2)*P(n-1,x) + (5*n-5+(48*n+42)*x+(117*n+207)*x^2+(54*n+108)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 33 + 36*x.
%t Mathematica
P[0, x_] := 33 + 36*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 6*n - 99 + (63*n^2 + 252*n + 189)*x + (54*n^2 + 270*n + 324)*x^2)*P[n - 1, x] + (5*n - 5 + (48*n + 42)*x + (117*n + 207)*x^2 + (54*n + 108)*x^3)*D[P[n - 1, x], x])/(6*n^2 - 12*n - 18 + (36*n^2 + 90*n + 54)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 33 + 36*x; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^2 - 6*n - 99 + (63*n^2 + 252*n + 189)*x + (54*n^2 + 270*n + 324)*x^2)*v[i - 1] + (5*n - 5 + (48*n + 42)*x + (117*n + 207)*x^2 + (54*n + 108)*x^3)*deriv(v[i - 1], x))/(6*n^2 - 12*n - 18 + (36*n^2 + 90*n + 54)*x)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023