Real-rooted atlas

A257609 → D (shift +2) → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-4 - 8\,n - 8\,n x - 2\,n x^{2} - 5\,n^{2} - 8\,n^{2} x - 3\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x + 6\,x^{2} + 2\,x^{3} + 4\,n x + 7\,n x^{2} + 3\,n x^{3} + n^{2} x + 2\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(6 + 2\,x + 7\,n + 4\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-x - x^{2} - n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 36 + 28\,x + 3\,x^{2}\)
\(P_{2}(x) = 240 + 300\,x + 90\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
13628367
2240300904634
318003120162024856793
41512033600252007224630681780
514112038304037800016329628980152471095967
614515204656960571536033364809324001089003570816205198
71632960060480000889056006574176025515000491112039186081769262283125
81995840008382528001437004800128997792064615320017764560024486000136804818414104614490792
9263450880012374208000242494560002564787456015810379200571817664011586960001174081924672404409401187715420747
10373621248001940754816004281076800005218243430403829617792001728344217604697941248072081609605470264801569750090090121791916217922
115666588928003225596774400791142992640010926345830400931717987200050591084544001740336998400365393508480432810378002494018800520720201965841339157877582497
12915372057600056665889280000152997901056000236224109721600229751121600000146157518707200611410415616001648045054656027260924090402529203040001118197080017100727242595814911562119166044
13156920924160000104962662604800030939575546880005282655733555200577534384627200042185886462720002084923320480000690847412935680148866194917440196914712176001446971526000494780834885570515089174761522522918738124407
142845499424768000204520271155200006535756491264000012230312215080960014854396215859200012263901387978240070066100744640000276625522985011207408556275760640129568848121440013864553919216081399623774402166332344801802457300196605016588721091381455606
1554420176498688000418288415440896000144053408378880000029322515884806144003918497243012352000361070122164111360023445944529815808001077380960066611200346760348504169600762193414120320001094732007969600095633784897696045174629050200940399146960579951090041942721716231597610587432909
Technical details
Node id
n-2422dc9bfda6e079 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-867bd83b37df747f

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:          36,           28,            3;
   2:         240,          300,           90,            4;
   3:        1800,         3120,         1620,          248,            5;
   4:       15120,        33600,        25200,         7224,          630,            6;
   5:      141120,       383040,       378000,       163296,        28980,         1524,           7;
   6:     1451520,      4656960,      5715360,      3336480,       932400,       108900,        3570,          8;
   7:    16329600,     60480000,     88905600,     65741760,     25515000,      4911120,      391860,       8176,         9;
   8:   199584000,    838252800,   1437004800,   1289977920,    646153200,    177645600,    24486000,    1368048,     18414,       10;
   9:  2634508800,  12374208000,  24249456000,  25647874560,  15810379200,   5718176640,  1158696000,  117408192,   4672404,    40940,    11;
  10: 37362124800, 194075481600, 428107680000, 521824343040, 382961779200, 172834421760, 46979412480, 7208160960, 547026480, 15697500, 90090, 12;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(n^3+5*n^2+8*n+4+(2*n^3+8*n^2+8*n)*x+(n^3+3*n^2+2*n)*x^2)*P(n-2,x) + ((n^2+4*n+4)*x+(2*n^2+7*n+6)*x^2+(n^2+3*n+2)*x^3)*d/dx P(n-2,x) + (2*n^2+7*n+6+(2*n^2+4*n+2)*x)*P(n-1,x) - ((n+1)*x+(n+1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 36 + 28*x + 3*x^2, P(2,x) = 240 + 300*x + 90*x^2 + 4*x^3.
%t Mathematica
P[1, x_] := 36 + 28*x + 3*x^2;
P[2, x_] := 240 + 300*x + 90*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 5*n^2 + 8*n + 4 + (2*n^3 + 8*n^2 + 8*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*P[n - 2, x] + ((n^2 + 4*n + 4)*x + (2*n^2 + 7*n + 6)*x^2 + (n^2 + 3*n + 2)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 7*n + 6 + (2*n^2 + 4*n + 2)*x)*P[n - 1, x] - ((n + 1)*x + (n + 1)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 36 + 28*x + 3*x^2; v[2] = 240 + 300*x + 90*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = (-(n^3 + 5*n^2 + 8*n + 4 + (2*n^3 + 8*n^2 + 8*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*v[i - 2] + ((n^2 + 4*n + 4)*x + (2*n^2 + 7*n + 6)*x^2 + (n^2 + 3*n + 2)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 7*n + 6 + (2*n^2 + 4*n + 2)*x)*v[i - 1] - ((n + 1)*x + (n + 1)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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

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.