Real-rooted atlas

A038319 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(21 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(416 + 210\,x + 25\,x^{2} + 80\,n + 25\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 416 + 210\,x + 25\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
041621025651
179366240157512515876
21474561587206240010500625379701
32686976368640019840005200006562531258946126
448234496806092805529600019840000390000039375015625208289151
585563801616882073601410662400645120000173600000273000002296875781254802902776
6150323855363422552064033764147200188088320006451200000138880000018200000013125000390625109866401001
7261993005056676457349120770074214400506462208000211599360000580608000001041600000011700000007382812519531252496308717826
8453548546457613099650252800169114337280001283457024000063307776000002115993600000483840000000744000000007312500000410156250976562556393883307251
97806532557209624945170055168036024038195200031004295168000017647534080000069638553600000193966080000003801600000000511500000000446875000002255859375488281251267670954343276
101337006139375616468391953432576074835510165504007204807639040000465064427520000021177040896000006963855360000001662566400000002851200000000034100000000002681250000001230468750024414062528372477543919901
112279947311356313686905399059415040152227384865587200162143605358592000117078124134400000604583755776000002294179430400000064664371200000001350835200000000205920000000000221650000000001584375000000666503906251220703125632601165979251126
1238730956795386265615959631179494195203041688967079526400355197231353036800028375130937753600001639093737881600000705347715072000000229417943040000000565813248000000001050649600000000014414400000000001410500000000009242187500000358886718750610351562514056986374259870951
1365572410574514421762904821759653969920059848616923103232000760422241769881600006659948087869440000042562696406630400000204886717235200000007557296947200000000215079321600000000047151104000000000078798720000000000982800000000000088156250000000053320312500000192260742187530517578125311416313522064833376
14110680464442257309696524579284596115374080116192870386158796800015959631179494195200001520844483539763200000106559169405911040000056750261875507200000023415624826880000000075572969472000000000191181619200000000003772088320000000000573081600000000000655200000000000005425000000000000304687500000000102539062500001525878906256880354176878119912401
Technical details
Node id
n-ffc8f2f3095cb69e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-723a18e430c13076

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            416,             210,              25;
  1:           7936,            6240,            1575,             125;
  2:         147456,          158720,           62400,           10500,             625;
  3:        2686976,         3686400,         1984000,          520000,           65625,           3125;
  4:       48234496,        80609280,        55296000,        19840000,         3900000,         393750,          15625;
  5:      855638016,      1688207360,      1410662400,       645120000,       173600000,       27300000,        2296875,         78125;
  6:    15032385536,     34225520640,     33764147200,     18808832000,      6451200000,     1388800000,      182000000,      13125000,       390625;
  7:   261993005056,    676457349120,    770074214400,    506462208000,    211599360000,    58060800000,    10416000000,    1170000000,     73828125,     1953125;
  8:  4535485464576,  13099650252800,  16911433728000,  12834570240000,   6330777600000,  2115993600000,   483840000000,   74400000000,   7312500000,   410156250,    9765625;
  9: 78065325572096, 249451700551680, 360240381952000, 310042951680000, 176475340800000, 69638553600000, 19396608000000, 3801600000000, 511500000000, 44687500000, 2255859375, 48828125;
%F Formula
The row polynomials satisfy (5*n+21+5*x)*P(n,x) = (80*n+416+(25*n+210)*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 416 + 210*x + 25*x^2.
%t Mathematica
P[0, x_] := 416 + 210*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((80*n + 416 + (25*n + 210)*x + 25*x^2)*P[n - 1, x])/(5*n + 21 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 416 + 210*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((80*n + 416 + (25*n + 210)*x + 25*x^2)*v[i - 1])/(5*n + 21 + 5*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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.