Real-rooted atlas

A038319 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(36 + 40\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1296 + 2880\,x + 1600\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01296288016005776
14665615552017280064000438976
216796167464960124416009216000256000033362176
3604661763359232007464960008294400004608000001024000002535525376
4217678233614511882240403107840005971968000049766400000221184000004096000000192699928576
5783641640966094990540802031663513600376233984000041803776000002786918400000103219200000016384000000014645194571776
6282110990745625076532510720975198486528002167107747840003009871872000002675441664000001486356480000004718592000000065536000000001113034787454976
710155995666841610155995666841604513775851929600117023818383360001950396973056000021671077478400000160526499840000007644119040000000212336640000000026214400000000084590643846578176
836561584400629764062398266736640020311991333683200060183678025728000011702381838336000001560317578444800000144473849856000000091729428480000000038220595200000000094371840000000000104857600000000006428888932339941376
91316217038422671361608709713627709440893727618682060800029790920622735360000662020458283008000001029809601773568000001144232890859520000009081213419520000000050451185664000000000186856243200000000004152360960000000000419430400000000000488595558857835544576
1047383813383216168966317841784428822528038609033127065026560014299641898912972800003574910474728243200000635539639951687680000082384768141885440000007846168394465280000000544872805171200000000026907299020800000000008969099673600000000001811939328000000000001677721600000000000037133262473195501387776
11170581728179578208256246395829592724078592016426388639514938572800669223240869127127040001858953446858686464000003717906893717372928000005508010212914626560000006120011347682918400000005100009456402432000000003148153985433600000000001399179549081600000000004239938027520000000000078517370880000000000006710886400000000000002822127947962858105470976
12614094221446481549721695525767780563796623360689908322859627420057600306625921270945520025600093691253721677797785600002082027860481728839680000034700464341362147328000000440640817033170124800000004284007943378042880000000031733392173170688000000000176296623184281600000000007123095886233600000000000197863774617600000000000033822867456000000000000026843545600000000000000214481724045177216015794176
132210739197207333578997763684565328678889298329600286577303341691389870080001379816645719254840115200004599388819064182800384000001124295044660133573427200000208202786048172883968000000029743255149738983424000000003304806127748775936000000000285600529558536192000000000019040035303902412800000000009616179446415360000000000003561547943116800000000000009132174213120000000000000014495514624000000000000000107374182400000000000000016300611027433468417200357376
14795866110994640088439193614148730862126934905585664011790609051772445754654720006113649137956082983895040000220770663315080774418432000005887217688402153984491520000011992480476374758116556800000019035683295832949391360000000023794604119791186739200000000023500843575102406656000000000018278433891746316288000000000011077838722270494720000000000051286290380881920000000000000175337744891904000000000000004174708211712000000000000000618475290624000000000000000429496729600000000000000001238846438084943599707227160576
Technical details
Node id
n-010e63e80d598424 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-249d425e0def4c4f

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:               1296,                2880,                1600;
  1:              46656,              155520,              172800,                64000;
  2:            1679616,             7464960,            12441600,              9216000,              2560000;
  3:           60466176,           335923200,           746496000,            829440000,            460800000,             102400000;
  4:         2176782336,         14511882240,         40310784000,          59719680000,          49766400000,           22118400000,            4096000000;
  5:        78364164096,        609499054080,       2031663513600,        3762339840000,        4180377600000,         2786918400000,         1032192000000,         163840000000;
  6:      2821109907456,      25076532510720,      97519848652800,      216710774784000,      300987187200000,       267544166400000,       148635648000000,       47185920000000,        6553600000000;
  7:    101559956668416,    1015599566684160,    4513775851929600,    11702381838336000,    19503969730560000,     21671077478400000,     16052649984000000,     7644119040000000,     2123366400000000,      262144000000000;
  8:   3656158440062976,   40623982667366400,  203119913336832000,   601836780257280000,  1170238183833600000,   1560317578444800000,   1444738498560000000,   917294284800000000,   382205952000000000,    94371840000000000,   10485760000000000;
  9: 131621703842267136, 1608709713627709440, 8937276186820608000, 29790920622735360000, 66202045828300800000, 102980960177356800000, 114423289085952000000, 90812134195200000000, 50451185664000000000, 18685624320000000000, 4152360960000000000, 419430400000000000;
%F Formula
The row polynomials satisfy P(n,x) = (36+40*x)*P(n-1,x) for n >= 1, with P(0,x) = 1296 + 2880*x + 1600*x^2.
%t Mathematica
P[0, x_] := 1296 + 2880*x + 1600*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 40*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1296 + 2880*x + 1600*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 40*x)*v[i - 1]); 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. [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.