Real-rooted atlas

A038281 → x ↦ x+1 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 252\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
019625281529
127445292340272912167
2384169878495256408246561279841
3537824172872022226401428840459270590496436343
47529536290424964667544040007520192893404960116531441148035889
51054135044743607689148386249801842406301184402430456845208121847829693404825447
614757890567589772288170769876482195612697617643316320907370553629165482085356925284304672178310985281
7206610467841195389135363073857776644610786664964446115712642858217243841224950247363374862926454238868463874204891801152661463
82892546549761859494210560537925110912092215733299201037426999616080030082827524287325865760157493603232037967207922054238868460348678440141426511213649
9404956516966428636210842624920449634227201775152866009602282339399155202054105459239681320496366654086063503724432019489833399960417639287142053696479775431381059609952809757913927
105669391237529643735303832371215463553855016963313618683217920479291273822592049298531021752323697389826631424203733725140915281857300279832023387800079952045105043011336527201801431228242953648121914624432020321
117937147732541446633187747909632255851527419371526030786003456614496923346484124160112154158074486528961321354924170246179922995941094429796057301858848106414490363781602736372609354384479753639302392514021756395422541865828329504036361936467383
1211112006825558016100008061430022144417890828118306816107457641516136038418996975910888335362442468331399928832235523731956421708817303784388635064329733378718607223684171448022260238721340822578583648163134390443442294450374132126751164982057023524842287679245496111592836324538749809
13155568095557812224150012092145033216067505441465264947201880508726532380672036266954011695912960512918349593985054725495553745649839872045422434020167043840292001361558216710401460006807791083552056314548300513222721645554982807204560352618924887258120523115987470108204804126415541810205891132094649266635235464391245607
14217795333780937113622401805760324960256108008706344423915520324026119033271746560676983141551657041920104448827553684229324812310040390255641313281130513913390824202240817603812363006789120467202178493146736640210240980321916031488737208632297627642881974665979368645472039059327064434745605380621585406827204611961358920137618530201888518416132610415680998648961
Technical details
Node id
n-ce03950bfb372f88 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-9111d60e4a92b32c

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:           196,            252,             81;
  1:          2744,           5292,           3402,             729;
  2:         38416,          98784,          95256,           40824,            6561;
  3:        537824,        1728720,        2222640,         1428840,          459270,           59049;
  4:       7529536,       29042496,       46675440,        40007520,        19289340,         4960116,          531441;
  5:     105413504,      474360768,      914838624,       980184240,       630118440,       243045684,        52081218,        4782969;
  6:    1475789056,     7589772288,    17076987648,     21956126976,     17643316320,      9073705536,      2916548208,      535692528,       43046721;
  7:   20661046784,   119538913536,   307385777664,    461078666496,    444611571264,    285821724384,    122495024736,    33748629264,     5423886846,     387420489;
  8:  289254654976,  1859494210560,  5379251109120,   9221573329920,  10374269996160,   8003008282752,   4287325865760,  1574936032320,   379672079220,   54238868460,   3486784401;
  9: 4049565169664, 28636210842624, 92044963422720, 177515286600960, 228233939915520, 205410545923968, 132049636665408, 60635037244320, 19489833399960, 4176392871420, 536964797754, 31381059609;
%F Formula
The row polynomials satisfy P(n,x) = (14+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 252*x + 81*x^2.
%t Mathematica
P[0, x_] := 196 + 252*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 9*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] = 196 + 252*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 9*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.