Real-rooted atlas

A123519 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(6 + 4\,n\right)\,P_{n}(x) = \left(48 + 30\,x + 3\,x^{2} + 32\,n + 24\,n x + 3\,n x^{2}\right)\,P_{n-1}(x) + \left(-16\,x - 16\,x^{2} - 3\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 64 + 56\,x + 11\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0645611131
1512704304411561
2409676805184148815318601
3327687782471680319366872571221651
42621447536648806405345281775363056821312641211
5209715270778881002700877209603486720922752132448795331472881
616777216650117121082654721011220485789696020794368457305656284829681375033361
7134217728587202560112407347212349603848576368643902218241162752002187225623563921107714468927451
81073741824523449139211324620800143109652481169267097664512000002433191936619284480101767104974888841340353252096051
98589934592461708984321114007142401591990026241496611553289714532352044412698624142968217603174999552463187264399463841542841634556225161
106871947673640372692582410748155658241713960386560182276063232013614962442247321902120962855741685768015409152015786211840207055302416237937657579617561422605881
1154975581388835046933135361020484229529617964774457344213265258905601802381348044811153845518336511620887347217391167078404326092390407653427200091180598406556348082148900390102515045411
12439804651110430236569763840955887921397761841338379141122413964893880322277826510192641595032574361608418355681689633645578944512101317665751042262557040640363295354880396467648002632071880801980511073668757939051
133518437208883225948474415513688510686035968018519211785584642657502424465408276988239124889621658301376757761293457670799360594745013305344210520239505408568884059176961152398176256016938212659201705255744001051423712029930320112793922580223201
14281474976710656221661544159641681143958129868801832556030014259228572115317817344326049349899386882816477417701376018783199266078720977228753220403239789584124477441263542626418688309613711196160573839401369607776194304000726569429760418193022081117014753152453402204739361
Technical details
Node id
n-2cbf726e15f9d5b1 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-4309d584dc07719c

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:         64,          56,           11;
  1:        512,         704,          304,           41;
  2:       4096,        7680,         5184,         1488,          153;
  3:      32768,       77824,        71680,        31936,         6872,         571;
  4:     262144,      753664,       880640,       534528,       177536,       30568,        2131;
  5:    2097152,     7077888,     10027008,      7720960,      3486720,      922752,      132448,        7953;
  6:   16777216,    65011712,    108265472,    101122048,     57896960,    20794368,     4573056,      562848,      29681;
  7:  134217728,   587202560,   1124073472,   1234960384,    857636864,   390221824,   116275200,    21872256,    2356392,    110771;
  8: 1073741824,  5234491392,  11324620800,  14310965248,  11692670976,  6451200000,  2433191936,   619284480,  101767104,   9748888,   413403;
  9: 8589934592, 46170898432, 111400714240, 159199002624, 149661155328, 97145323520, 44412698624, 14296821760, 3174999552, 463187264, 39946384, 1542841;
%F Formula
The row polynomials satisfy (4*n+6)*P(n,x) = (32*n+48+(24*n+30)*x+(3*n+3)*x^2)*P(n-1,x) - (16*x+16*x^2+3*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 64 + 56*x + 11*x^2.
%t Mathematica
P[0, x_] := 64 + 56*x + 11*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((32*n + 48 + (24*n + 30)*x + (3*n + 3)*x^2)*P[n - 1, x] - (16*x + 16*x^2 + 3*x^3)*D[P[n - 1, x], x])/(4*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 64 + 56*x + 11*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((32*n + 48 + (24*n + 30)*x + (3*n + 3)*x^2)*v[i - 1] - (16*x + 16*x^2 + 3*x^3)*deriv(v[i - 1], x))/(4*n + 6)); 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

Same fingerprint ?

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023