Real-rooted atlas

A038301 → x ↦ x+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(31 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 961 + 682\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
09616821211764
1297913171311253133174088
29235211310804697686165044146413111696
3286291515079365536047110127909102269355161051130691232
4887503681188952396616761906157930364202110500152995548617715615489031744
52751261411168337783437727466726914302222578515265951085325017023138442873719487171230539333248
68528910374412421110041768300686247122821339023989369464889672702686807390964766916338848328184082143588819682651996416
726439622160671844362127066591198449470675169922646155052452814084373666187404815519464433232195084674178167916598061277992357947691406671383849472
8819628286980801290835843767381046439916988662454394314725808920272872769263941011619098562206523435754951190106966507735132092699498088457309637842102593742460117080198121677824
925408476896404831991750227246769211759556854792655051873076651876052151329280204557198306603521016187372223431848767116482593894784420003010536842949387151246293252078058844661788941285311670611717368321110468608
1078766278378854976133539189503254376926545551499828676786774205016108768222061811529511909720953509299740031003512145277462356122188444185771960848223297992639429300495154540363257678201645107092743026106135941467292313842837672130129469486639681536
1124417546297445042591112635778081762615823239805204948268794978312004621491833593466276778293258884639365176780974404061801917836383104707389170362967811016703638991678982067380016198971556994159821107865220992719458479826213863922056593381264786635818563345227121439311265437718438866624512
127569439352207963203213760302129806536559014867295491229572141837112310000520677798142204120121779274355933484418524771432373646892442453737834303758624920318400428303562561747925713036207154505058831351470930724721626822397771006124506112130938980724187175642744586999726281711498285707046605437974983358324153148384174432398229504
13234652619918446859299511248957493114313928529653102249257090392661186554770125201762646780104055077875214779591733659153964018716053745804985532344312143902752895421551069524895144573901597853795088337609778360508510473899713116592607855222992703569579068425335966565091867591681542541098495757366505348351426888335755517658367261620706541772481694156512232232135326160725639168
1472742312174718526382848141298861105646647236713761099082593940596257106092018199862308263636945627760209885508877556458324578201787412075602416290248020811627788233757654361290888589427053324120727991704023529547693180625835152707421506082436899904994401843406349508520298982483567883257113265094804852750962134739134466205759410257887151157604379275080881935212020719150920301628964594972986357216193753749683698750476845056
Technical details
Node id
n-3b96364a23191cc0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5e64a37ca1ebb259

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:               961,               682,                121;
  1:             29791,             31713,              11253,               1331;
  2:            923521,           1310804,             697686,             165044,              14641;
  3:          28629151,          50793655,           36047110,           12790910,            2269355,            161051;
  4:         887503681,        1889523966,         1676190615,          793036420,          211050015,          29955486,           1771561;
  5:       27512614111,       68337783437,        72746672691,        43022225785,        15265951085,        3250170231,         384428737,         19487171;
  6:      852891037441,     2421110041768,      3006862471228,      2133902398936,       946488967270,      268680739096,       47669163388,       4832818408,        214358881;
  7:    26439622160671,    84436212706659,    119844947067516,     99226461550524,     52814084373666,    18740481551946,     4433232195084,     674178167916,      59806127799,      2357947691;
  8:   819628286980801,  2908358437673810,   4643991698866245,   4394314725808920,   2728727692639410,  1161909856220652,   343575495119010,   69665077351320,    9269949808845,    730963784210,   25937424601;
  9: 25408476896404831, 99175022724676921, 175955685479265505, 187307665187605215, 132928020455719830, 66035210161873722, 23431848767116482, 5938947844200030, 1053684294938715, 124629325207805, 8844661788941, 285311670611;
%F Formula
The row polynomials satisfy P(n,x) = (31+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 961 + 682*x + 121*x^2.
%t Mathematica
P[0, x_] := 961 + 682*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(31 + 11*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] = 961 + 682*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (31 + 11*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.