Real-rooted atlas

A257616 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-8 + 6\,x + 3\,n\right)\,P_{n}(x) = \left(80 - 476\,x + 1616\,x^{2} - 1692\,x^{3} + 360\,x^{4} - 48\,n + 348\,n x - 1800\,n x^{2} + 1512\,n x^{3} - 216\,n x^{4} - 72\,n^{2} x + 756\,n^{2} x^{2} - 324\,n^{2} x^{3} - 108\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(60\,x - 612\,x^{2} + 1308\,x^{3} - 972\,x^{4} + 216\,x^{5} - 36\,n x + 504\,n x^{2} - 792\,n x^{3} + 324\,n x^{4} - 108\,n^{2} x^{2} + 108\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-74 + 268\,x - 222\,x^{2} + 36\,x^{3} + 30\,n - 174\,n x + 90\,n x^{2} + 36\,n^{2} x\right)\,P_{n-1}(x) + \left(-48\,x + 84\,x^{2} - 36\,x^{3} + 18\,n x - 18\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
23640480
332093633681600
42672201281670427201642240
52169637251270169626208021824321379840
61744006212736240700801909376038908801747206453616640
713972489709222470436275210941920004580761605623027213980161282412748800
81118284814570951681844662528051426276864409190451201024562739280123443211184640256123318272000
9894740482131631769644750381875220853180764162909531916288135870277939221992408678411329210368894781445127053805158400
1071581798430683898880010291841233920759472598630401733169521971201398865070776324179906917376046040859033601595039385607158272001024446313490022400
11572660121643716225044482279177788006402556812682455040903295723757568011777591424012288602525660354150412223514836377609489147518976022402139443205726621696204830944401974886400
124581293670461878453903360491234819348889681196893890682880425580752811110400848363618686943232695834634062561280240277162931601408345367331022643201936307162521600314202627440644581298176040962332731841183744000
133665037721608722540361891841038150168365629442467753435799437312185634588974988492805416595645518393344067433239150563950592371573716319082577929065938018006941696952920346428497920392550724517888004403418113802243665038704648192189951021353533440000
14293203078348812263600041787392216264432371988889672530478734562754560762847807080572207104314813442284857861734456922186551716514856964740639555282021777408184067005499116118016032850808122419311411225860128146783879168792402004637581312616845040548249629320309964801638416614382667722391552000
Technical details
Node id
n-401b73bdd722880d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2b6b8baf2ca9e23d

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:        4,           2;
  2:       36,          40,            4;
  3:      320,         936,          336,             8;
  4:     2672,       20128,        16704,          2720,            16;
  5:    21696,      372512,       701696,        262080,         21824,            32;
  6:   174400,     6212736,     24070080,      19093760,       3890880,        174720,           64;
  7:  1397248,    97092224,    704362752,    1094192000,     458076160,      56230272,      1398016,         128;
  8: 11182848,  1457095168,  18446625280,   51426276864,   40919045120,   10245627392,    801234432,    11184640,      256;
  9: 89474048, 21316317696, 447503818752, 2085318076416, 2909531916288, 1358702779392, 219924086784, 11329210368, 89478144, 512;
%F Formula
The row polynomials satisfy (3*n-8+6*x)*P(n,x) = -(48*n-80+(72*n^2-348*n+476)*x+(108*n^3-756*n^2+1800*n-1616)*x^2+(324*n^2-1512*n+1692)*x^3+(216*n-360)*x^4)*P(n-2,x) - ((36*n-60)*x+(108*n^2-504*n+612)*x^2-(108*n^2-792*n+1308)*x^3-(324*n-972)*x^4-216*x^5)*d/dx P(n-2,x) + (30*n-74+(36*n^2-174*n+268)*x+(90*n-222)*x^2+36*x^3)*P(n-1,x) + ((18*n-48)*x-(18*n-84)*x^2-36*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(48*n - 80 + (72*n^2 - 348*n + 476)*x + (108*n^3 - 756*n^2 + 1800*n - 1616)*x^2 + (324*n^2 - 1512*n + 1692)*x^3 + (216*n - 360)*x^4)*P[n - 2, x] - ((36*n - 60)*x + (108*n^2 - 504*n + 612)*x^2 - (108*n^2 - 792*n + 1308)*x^3 - (324*n - 972)*x^4 - 216*x^5)*D[P[n - 2, x], x] + (30*n - 74 + (36*n^2 - 174*n + 268)*x + (90*n - 222)*x^2 + 36*x^3)*P[n - 1, x] + ((18*n - 48)*x - (18*n - 84)*x^2 - 36*x^3)*D[P[n - 1, x], x])/(3*n - 8 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(48*n - 80 + (72*n^2 - 348*n + 476)*x + (108*n^3 - 756*n^2 + 1800*n - 1616)*x^2 + (324*n^2 - 1512*n + 1692)*x^3 + (216*n - 360)*x^4)*v[i - 2] - ((36*n - 60)*x + (108*n^2 - 504*n + 612)*x^2 - (108*n^2 - 792*n + 1308)*x^3 - (324*n - 972)*x^4 - 216*x^5)*deriv(v[i - 2], x) + (30*n - 74 + (36*n^2 - 174*n + 268)*x + (90*n - 222)*x^2 + 36*x^3)*v[i - 1] + ((18*n - 48)*x - (18*n - 84)*x^2 - 36*x^3)*deriv(v[i - 1], x))/(3*n - 8 + 6*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

Same fingerprint ?

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.