Real-rooted atlas

A038226 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 16\,x + 8\,n x\right)\,P_{n}(x) = \left(121 + 352\,x + 192\,x^{2} + 88\,n x + 64\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 352\,x + 192\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0121352192665
1133158086336204815523
214641851841393929011220480349809
316105111712802555520247808011264001966087688939
4177156115460896421660805451776037171200129761281835008165898633
5194871711984148326493576321049466880954060800499580928141295616167772163528441075
6214358881249435788895239119361847061708820989337600146543738886217007104147639500815099494474191354337
7235794769130867678864134695325952304765181952415588884480362695753728205161234432730815528961494849945613421772801545504236731
8259374246013772716305601852060731840478916714496076191295488007979306582016564193394688026796569395208221674700801476395008001181116006431946082080121
928531167061145649867297762489992761696072436153067520131702096486400160916016070656136534801514496810596224204803316075462656089321897984001429150367744103079215104656024089584707
10313842837672154779840757312328679044543872106239691165696021730845920256003034416303046656300376563331891221399740319006721094304902676480393016351129600943239242711041360645639372889335319756813396379773295185
113452271214393165279310235796842728275790703361519227583669452834527899628851200542401914169589766136264079494348851002714426966016312971202165473281405033455288320044961070569226249728616321515521277495072522247696581394432272237735045137803
12379749833583241773308752024054454834620598069312212691861713723392531729654284308480928109942023520256118123083530266214411220597173932523528032927522247147524327503042288025601731001216915210244994023045044633698367120584212481185273534742528659706976665605508940833555674729
13417724816941565191139960059977840695977876821648960292451309856369664079759448142646272001531381404338808422421655898647215472640231424816712358297601893475773101113344011900633366292070400571230401582019379220600345060809113605410191632131686409778506661625856010885165114982400562949953421312111061929323820892819
14459497298635721611069375531370406656874943616575787264039593408103631585280116980523942547865600245021024694209347584381143816190992318464452564086015278448640416564670082244935680299215924638200627200167560917797392351232725132146140480798722380484318137942016057367239081538355209578945301184512009907919180215091247850746040811522231625673391595349953
Technical details
Node id
n-ccee011b772bff0f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d0117ab4ab9edada

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:          121,           352,            192;
  1:         1331,          5808,           6336,           2048;
  2:        14641,         85184,         139392,          90112,           20480;
  3:       161051,       1171280,        2555520,        2478080,         1126400,          196608;
  4:      1771561,      15460896,       42166080,       54517760,        37171200,        12976128,         1835008;
  5:     19487171,     198414832,      649357632,     1049466880,       954060800,       499580928,       141295616,       16777216;
  6:    214358881,    2494357888,     9523911936,    18470617088,     20989337600,     14654373888,      6217007104,     1476395008,      150994944;
  7:   2357947691,   30867678864,   134695325952,   304765181952,    415588884480,    362695753728,    205161234432,    73081552896,    14948499456,    1342177280;
  8:  25937424601,  377271630560,  1852060731840,  4789167144960,   7619129548800,   7979306582016,   5641933946880,  2679656939520,   822167470080,  147639500800,   11811160064;
  9: 285311670611, 4564986729776, 24899927616960, 72436153067520, 131702096486400, 160916016070656, 136534801514496, 81059622420480, 33160754626560, 8932189798400, 1429150367744, 103079215104;
%F Formula
The row polynomials satisfy (11+(8*n+16)*x)*P(n,x) = (121+(88*n+352)*x+(64*n+192)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 352*x + 192*x^2.
%t Mathematica
P[0, x_] := 121 + 352*x + 192*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((121 + (88*n + 352)*x + (64*n + 192)*x^2)*P[n - 1, x])/(11 + (8*n + 16)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 352*x + 192*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((121 + (88*n + 352)*x + (64*n + 192)*x^2)*v[i - 1])/(11 + (8*n + 16)*x)); 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.