Real-rooted atlas

A038320 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(35 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1225 + 1260\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0122512603242809
14287566150340205832148877
21500625308700023814008164801049767890481
35252187513505625013891500071442000183708001889568418195493
4183826562556723625007293037500500094000019289340003968092803401222422164361129
5643392968752316214687503573588375003063075750001575296100004860913680083329948806122200321174711139837
622518753906259264858750000166767457500001715322420000011027072700000453685276800011666192832001714216089601101996057662259690411361
7788156386718753648038132812507504535587500009005442705000006947055801000003572771554800001224950247360002699890341120034712875814401983592903683299763591802133
827585473535156251418681496093750032832343195312500450272135250000004052449217250000025009400883600000107183146644000003149872064640000607475326752000694257516288003570467226624174887470365513049
996549157373046875546192375996093750140449468113281250021669346508906250002228847069487500000160476989003100000082531022915880000030317518622160000077959333599840000133644571885440001374629882250240642684100792329269035929372191597
103379220508056640625208546179925781250005898877660757812500010112361704156250000011701447114809375000096286193401860000000577717160411160000002546671564261440000081857300279832000001871024006396160000288672275272550400269927322332774401156831381426176491258904256726154641
111182727177819824218757907375988852539062502439990305131640625000460112457539109375000059157315969314062500005476277249730787500000375516154267254000000019312259362315920000007449014325464712000002128289807275632000004378196174967014400061408465830706176005263582785489100802082296486567116826036721925606486195973
1241395451223693847656252980472488105957031250099632937459541992187500204959185631057812500000289870848249638906250000298152872485342875000000230003644488693075000000135185815536211440000000608336169912951480000002085724011130119360000053632903143345926400001003004941901534208000128957778244482969600102032527841788723203748133675820810241379946262057143768386569
13144884079282928466796875111767718303973388671875040236378589430419921875008966964371358779296875000138347450300964023437500001565302580548050093750000013416879261840429375000000887156914456387575000000045625212743471361000000001825008509738854440000000563145483005132227200000131644398624576364800000225676111927845196800002678353855846953984000196777017980592537600674664061647745843273137151889028619724488157
14507094277490249633789062541726614833483398437500000160945514357721679687500000386269234458532031250000000645621434737832109375000000796881313733552775000000000751345238663064045000000000552008746772863380000000000319376489204299527000000000146000680779108355200000000525602450804790078720000001474417264595255285760000031594655669898327552000004999593864247647436800005509756503456591052800037781187452273767219201214395310965942517763876269050118516845397872321
Technical details
Node id
n-afcbcd66bd66811b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-fbd68fd8efed4388

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:              1225,               1260,                 324;
  1:             42875,              66150,               34020,                5832;
  2:           1500625,            3087000,             2381400,              816480,              104976;
  3:          52521875,          135056250,           138915000,            71442000,            18370800,             1889568;
  4:        1838265625,         5672362500,          7293037500,          5000940000,          1928934000,           396809280,           34012224;
  5:       64339296875,       231621468750,        357358837500,        306307575000,        157529610000,         48609136800,         8332994880,          612220032;
  6:     2251875390625,      9264858750000,      16676745750000,      17153224200000,      11027072700000,       4536852768000,      1166619283200,       171421608960,       11019960576;
  7:    78815638671875,    364803813281250,     750453558750000,     900544270500000,     694705580100000,     357277155480000,    122495024736000,     26998903411200,     3471287581440,      198359290368;
  8:  2758547353515625,  14186814960937500,   32832343195312500,   45027213525000000,   40524492172500000,   25009400883600000,  10718314664400000,   3149872064640000,   607475326752000,    69425751628800,    3570467226624;
  9: 96549157373046875, 546192375996093750, 1404494681132812500, 2166934650890625000, 2228847069487500000, 1604769890031000000, 825310229158800000, 303175186221600000, 77959333599840000, 13364457188544000, 1374629882250240, 64268410079232;
%F Formula
The row polynomials satisfy P(n,x) = (35+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 1225 + 1260*x + 324*x^2.
%t Mathematica
P[0, x_] := 1225 + 1260*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(35 + 18*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] = 1225 + 1260*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (35 + 18*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

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.