Real-rooted atlas

A038333 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(144 + 420\,x + 196\,x^{2} + 252\,n x + 196\,n x^{2} + 49\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1728 + 9072\,x + 9996\,x^{2} + 3087\,x^{3} + 3024\,n x + 5292\,n x^{2} + 2058\,n x^{3} + 588\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 672\,x + 441\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01446724411257
117281209615876548835188
22073619353638102426342460025918745
3248832290304076204807902720360150060505222881624
4298598441803776137168640189665280129654000435637445764801550606225
53583180858525286423044331523982970880363031200018296772484842432845270675212905427988
6429981696802632499236870930432764730408968712748800058549671936232436776325059848192466948881296247912657
751597803521083553873925688657838081376514736128188195374080015808411422728367723947522732318023685043047914840353607006686160607720
861917364224144473849856085329867571202359739547648037639074816000379401874145282510317184256010929272094720302582874888048424328400034179505129148792995802201
9743008370688190705481809921251504724377603893570253619207097654108160008346841231196166627237366435843606659791257601331364649507203196005674400045116946770282847350509923272053255479060
10891610044825624965081255116818021668031037446229712405790720127757773946880001717064481846067215905369679446016103871801988218884792912738225920153408272371200032484201674601641001847342848233917753696971224596716874217
111069932053790723245460563165184255580019349258249718351353033523222144680817459200033482757395998310435446252428479692827006668516936908814953887743264870459829226224768000168917848707928323198144092742144364911695767164189902460397721536739495385305048
12128391846454886441941336508596224357812027088961536148425729755421081637203063773331456006250114713919684608744371300998073548864816004440648581124187088568114163712201026200115220480070945496457329894417909606919356006430652582444441776319036133468169615260019139102532900926271918483265
131540702157458636853924575511052288049543203750779289602226385946331316224060877740719996928000112502064850554322944148874260199614709760145836009991459307520107667991751507066880603078600345661440002554037872463876198480593231137102028801839154946666506560287132520121352640274680344503845001215375746545408699593432878108433524
141848842588950364166902345665414692864679449651439258828803288200782273943961609740438515199508480001963672404664220909568285838579583260242739231111682131511318937602584031802036169605120165415844666238566400081729211918844038348830947800756647179059288279437439992314880183764812877665689602636931307236912000233352143336718336960431693461468914785618720412270659105
Technical details
Node id
n-6ed674fe0b3aea00 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-576b9a231d8a9de1

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          144,            672,             441;
  1:         1728,          12096,           15876,            5488;
  2:        20736,         193536,          381024,          263424,           60025;
  3:       248832,        2903040,         7620480,         7902720,         3601500,          605052;
  4:      2985984,       41803776,       137168640,       189665280,       129654000,        43563744,         5764801;
  5:     35831808,      585252864,      2304433152,      3982970880,      3630312000,      1829677248,       484243284,        52706752;
  6:    429981696,     8026324992,     36870930432,     76473040896,     87127488000,     58549671936,     23243677632,      5059848192,       466948881;
  7:   5159780352,   108355387392,    568865783808,   1376514736128,   1881953740800,   1580841142272,    836772394752,    273231802368,     50430479148,     4035360700;
  8:  61917364224,  1444738498560,   8532986757120,  23597395476480,  37639074816000,  37940187414528,  25103171842560,  10929272094720,   3025828748880,   484243284000,   34179505129;
  9: 743008370688, 19070548180992, 125150472437760, 389357025361920, 709765410816000, 834684123119616, 662723736643584, 360665979125760, 133136464950720, 31960056744000, 4511694677028, 284735050992;
%F Formula
The row polynomials satisfy (144+(252*n+420)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (1728+(3024*n+9072)*x+(588*n^2+5292*n+9996)*x^2+(343*n^2+2058*n+3087)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 672*x + 441*x^2.
%t Mathematica
P[0, x_] := 144 + 672*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1728 + (3024*n + 9072)*x + (588*n^2 + 5292*n + 9996)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*P[n - 1, x])/(144 + (252*n + 420)*x + (49*n^2 + 196*n + 196)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 672*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1728 + (3024*n + 9072)*x + (588*n^2 + 5292*n + 9996)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*v[i - 1])/(144 + (252*n + 420)*x + (49*n^2 + 196*n + 196)*x^2)); 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.