Real-rooted atlas

A139548 → x ↦ x+1 → basis A013610

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 36\,x + 18\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
118361872
23241296194412963245184
35832349928748011664087480349925832373248
41049768398082939328587865673483205878656293932883980810497626873856
518895681889568085030560226748160396809280476171136396809280226748160850305601889568018895681934917632
6340122244081466882244806784748268928016836050880269376814083142729497626937681408168360508807482689280224480678440814668834012224139314069504
76122200328571080448557120229122228480916486128322520321225664504064183849675609621011391498241838496756096122566450406461283225203222284809164855712022912857108044861222003210030613004288
811019960576176319369216132239526912061711779225602005632824832048135187795968882478442926081260683489894401418268926131201260683489894408824784429260848135187795968200563282483206171177922560132239526912017631936921611019960576722204136308736
919835929036835704672266243034897142630416186118094028860697942852608016995423998730243682341866391552631258605667123286798058279229449644228697692160867980582792294463125860566712323682341866391552169954239987302460697942852608016186118094028830348971426304357046722662419835929036851998697814228992
1035704672266247140934453248067838877305856040703326383513601729891371299328055356523881578496138391309703946240276782619407892480449771756537825280599695675383767040659665242922143744599695675383767040449771756537825280276782619407892480138391309703946240553565238815784961729891371299328040703326383513606783887730585607140934453248035704672266243743906242624487424
11642684100792321413905021743104148460027283025929897335152201728047012341972958208016924443110264954884795258881241737216109605917285525422082055110949103601664031968392541611581440415589103040950558724533699305901278822441558910304095055872319683925416115814402055110949103601664010960591728552542208479525888124173721616924443110264954884701234197295820809897335152201728014846002728302592141390502174310464268410079232269561249468963094528
121156831381426176277639531542282243192854612736245762341426716006580224122924902590345461764916996103613818470415570487661443758489640038396843712521830485081593292889108889615125616585402508247042268842487810376237056288761771175866066534431282525210718823874562887617711758660665344226884248781037623705615125616585402508247048508159329288910888964003839684371252183041557048766144375848964916996103613818470412292490259034546176234142671600658022431928546127362457627763953154228224115683138142617619408409961765342806016
13208229648656711685413970865074503686767463581343129600541397086507450368003113033247417839616001369734628863849431040479407120102347300864013697346288638494310400325311974355164239872006506239487103284797440011060607128075584155648016088155822655395135488020110194778319243919360021657132838189954990080020110194778319243919360016088155822655395135488011060607128075584155648065062394871032847974400325311974355164239872001369734628863849431040047940712010234730086401369734628863849431040311303324741783961600541397086507450368006767463581343129600541397086507450368208229648656711681397405517247104682033152
14374813367582081024104947742922982686721416794529460266270721227888592198897434624767430370124310896640036836657765966923038720141207188102873204981760443794019751887215656960116495930184870394109952025887984485526754246656004918717052250083306864640804880972186377268396032011402480439307011302277120140338220791470908335718401503623794194331160739840014033822079147090833571840114024804393070113022771208048809721863772683960320491871705225008330686464025887984485526754246656001164959301848703941099520443794019751887215656960141207188102873204981760368366577659669230387207674303701243108966400122788859219889743462414167945294602662707210494774292298268672374813367582081024100613197241791537106386944
Technical details
Node id
n-8f11dd840e2cc46c · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-2cd9f46ca2d53115

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • 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:            1;
  1:           18,            36,             18;
  2:          324,          1296,           1944,            1296,             324;
  3:         5832,         34992,          87480,          116640,           87480,            34992,             5832;
  4:       104976,        839808,        2939328,         5878656,         7348320,          5878656,          2939328,           839808,           104976;
  5:      1889568,      18895680,       85030560,       226748160,       396809280,        476171136,        396809280,        226748160,         85030560,         18895680,          1889568;
  6:     34012224,     408146688,     2244806784,      7482689280,     16836050880,      26937681408,      31427294976,      26937681408,      16836050880,       7482689280,       2244806784,        408146688,         34012224;
  7:    612220032,    8571080448,    55712022912,    222848091648,    612832252032,    1225664504064,    1838496756096,    2101139149824,    1838496756096,    1225664504064,     612832252032,     222848091648,      55712022912,       8571080448,       612220032;
  8:  11019960576,  176319369216,  1322395269120,   6171177922560,  20056328248320,   48135187795968,   88247844292608,  126068348989440,  141826892613120,  126068348989440,   88247844292608,   48135187795968,   20056328248320,    6171177922560,   1322395269120,    176319369216,    11019960576;
  9: 198359290368, 3570467226624, 30348971426304, 161861180940288, 606979428526080, 1699542399873024, 3682341866391552, 6312586056671232, 8679805827922944, 9644228697692160, 8679805827922944, 6312586056671232, 3682341866391552, 1699542399873024, 606979428526080, 161861180940288, 30348971426304, 3570467226624, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (18+36*x+18*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(18 + 36*x + 18*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (18 + 36*x + 18*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.