Real-rooted atlas

A166343 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,x + n\right)\,P_{n}(x) = \left(-4\,x + 6\,x^{2} - 2\,x^{3} - 2\,n + 3\,n x - 5\,n x^{2} + 4\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x + 2\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + 4\,x^{3} - 4\,x^{4} + 2\,x^{5} - n x + 2\,n x^{2} - 4\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 7\,x - 4\,x^{2} + 2\,x^{3} + 3\,n - 3\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + 3\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 13 + 14\,x + x^{2}\)
\(P_{2}(x) = 28 + 81\,x + 30\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
11314128
22881301140
359382336621840
412215572872120212615880
5249576619991167803999254147040
65042003312007818385585596127835101423360
71015666466498381676514139608040257039914102214233600
8203821515732637841334278418435564938704217983281228562046146569600
94085680110155243499598141620708546617325008458164610776621637476940941558835200
108180211793770926574639874669205582377626623488501459357732340470546327844761136669819017264857600
1116371652558231438775640250919101855625275535463367708299309682561135794322819137446651362571103434676163821101708006400
1232754199532691361894856242059911661555098164904222802117675231546877123047079038288336372515810445285235560207704103532943276611525620096000
1365521606786465796614243140527208164122901898464146008022352188051153743563686221430018428768499904275853863056585579440456122857813163115832365534124409921536000
1413105618380499324339023646793368846459926834806422046680318644169111871228531874134220312393723822122380745522562170512863539739430302742933103594339276276156936717391310701414968666112000
152621275552155901011233761384380332123370672981346783624470544337313181430886290017786234160846490599420151918242452809171037043344662179672258122382628244156519815236781033183749997405028140862226214217469435990016000
Technical details
Node id
n-f3d0389fa7364339 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-61e8cab8d65b78d2

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 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   13,      14,        1;
   2:   28,      81,       30,         1;
   3:   59,     382,      336,        62,          1;
   4:  122,    1557,     2872,      1202,        126,          1;
   5:  249,    5766,    19991,     16780,       3999,        254,          1;
   6:  504,   20033,   120078,    183855,      85596,      12783,        510,         1;
   7: 1015,   66646,   649838,   1676514,    1396080,     402570,      39914,      1022,        1;
   8: 2038,  215157,  3263784,  13342784,   18435564,    9387042,    1798328,    122856,     2046,       1;
   9: 4085,  680110, 15524349,  95981416,  207085466,  173250084,   58164610,   7766216,   374769,    4094,    1;
  10: 8180, 2117937, 70926574, 639874669, 2055823776, 2662348850, 1459357732, 340470546, 32784476, 1136669, 8190, 1;
%F Formula
The row polynomials satisfy (n-1+2*x)*P(n,x) = -(2*n+(2*n^2-3*n+4)*x+(n^3-2*n^2+5*n-6)*x^2+(3*n^2-4*n+2)*x^3+2*n*x^4)*P(n-2,x) - (n*x+(n^2-2*n+2)*x^2-(n^2-4*n+4)*x^3-(3*n-4)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n-2+(2*n^2-3*n+7)*x+(5*n-4)*x^2+2*x^3)*P(n-1,x) + ((n-1)*x-(n-3)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 13 + 14*x + x^2, P(2,x) = 28 + 81*x + 30*x^2 + x^3.
%t Mathematica
P[1, x_] := 13 + 14*x + x^2;
P[2, x_] := 28 + 81*x + 30*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (2*n^2 - 3*n + 4)*x + (n^3 - 2*n^2 + 5*n - 6)*x^2 + (3*n^2 - 4*n + 2)*x^3 + 2*n*x^4)*P[n - 2, x] - (n*x + (n^2 - 2*n + 2)*x^2 - (n^2 - 4*n + 4)*x^3 - (3*n - 4)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n - 2 + (2*n^2 - 3*n + 7)*x + (5*n - 4)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 1)*x - (n - 3)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 13 + 14*x + x^2; v[2] = 28 + 81*x + 30*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n + (2*n^2 - 3*n + 4)*x + (n^3 - 2*n^2 + 5*n - 6)*x^2 + (3*n^2 - 4*n + 2)*x^3 + 2*n*x^4)*v[i - 2] - (n*x + (n^2 - 2*n + 2)*x^2 - (n^2 - 4*n + 4)*x^3 - (3*n - 4)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n - 2 + (2*n^2 - 3*n + 7)*x + (5*n - 4)*x^2 + 2*x^3)*v[i - 1] + ((n - 1)*x - (n - 3)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 1 + 2*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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.