Real-rooted atlas

A132166 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0)coefficients of one signHadamard kernel

Recurrence

\(\left(3 + 6\,x + 3\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(48 + 110\,x + 39\,x^{2} - 60\,x^{3} - 37\,x^{4} + 6\,n - 18\,n x - 91\,n x^{2} - 104\,n x^{3} - 37\,n x^{4}\right)\,P_{n-1}(x) + \left(36\,x + 145\,x^{2} + 219\,x^{3} + 147\,x^{4} + 37\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 336 + 714\,x + 379\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
03367143791429
145361472115897571340867
254432240912398475292026800321065877
35987523406032770181386607574846548108032526294227
46158592434833921265561281947022921672312387611483014354557628601029
5600462725149422721859594760367947000943194147663014218107115942802718985343814796967651
656043187257583768322518390742461661018016928681607258838203104852022530263173447820662511177679346292415925
7504388684861571083392320562481824945826659648175585255758921371203479601711096972029871096234389256316427399332580003018097744651379
84401937612863522405273638867733515521355259993984030159831310128450689060359984600811458959331781839649040142505610024423751683617316440953665088189580506589861
937416469708863659965824004531378269600018391427576121648061561603816885708850374145710706340538804959405417062005385710739118498182287180857278465446701746947258499786175854444957093262083
1031084451758086228403111526451140296780876823866257885765127207953009834528150220428575729762231661389992076723927518660878882184384502322890319983543557538124361198228964722878512358737290877627638378537104334276966109333
1125311625003008597166856552448561469877095449629824487133755904102780719782581600246338810021031984426214899720279189542099258968614543508845346176368727349121842013754675170541237431036412562571687562431221124665629795660910301360542273662450537739608360083
12202493000024064562668049140940860188048037199872360793415199744000140408118378907084838210949265707002887577510933920916160111970938229576552081244437466281516868410384616983536693612642001985270313114628564334422469160848658402028820075461602278233227135301366971717831381157571774188651664261
1315946323751895045221155447495475263171612177850368042422484244885125121848336625676078611256529331756628533056126872916075419963112214367658277065454665276289721254010756556272591267019439657050204722240352660324630115161946422219393957470420609279706230581318945727533049528122718930983114997011813879896406450811352631017438575292707
1412381851383824384477889435733262336650580119026027315248638108529223852032235544304524186168064802734673783792537728201831424268957278588838506624319325891870725664800352509502516297647225654670982492358457402772041529086876183918838370992834195548202220003692672769112376349852534447174707219912532635446079929332079943483761215262240684001322817481631778372980424461841269
Technical details
Node id
n-c0d9843b43e10688 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-935f97559657f6f5

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          336,           714,            379;
  1:         4536,         14721,          15897,            5713;
  2:        54432,        240912,         398475,          292026,           80032;
  3:       598752,       3406032,        7701813,         8660757,         4846548,         1080325;
  4:      6158592,      43483392,      126556128,       194702292,       167231238,        76114830,         14354557;
  5:     60046272,     514942272,     1859594760,      3679470009,      4319414766,      3014218107,       1159428027,       189853438;
  6:    560431872,    5758376832,    25183907424,     61661018016,     92868160725,     88382031048,      52022530263,     17344782066,      2511177679;
  7:   5043886848,   61571083392,   320562481824,    945826659648,   1755852557589,   2137120347960,    1711096972029,    871096234389,    256316427399,     33258000301;
  8:  44019376128,  635224052736,  3886773351552,  13552599939840,  30159831310128,  45068906035998,   46008114589593,  31781839649040,  14250561002442,   3751683617316,   440953665088;
  9: 374164697088, 6365996582400, 45313782696000, 183914275761216, 480615616038168, 857088503741457, 1070634053880495, 940541706200538, 571073911849818, 228718085727846, 54467017469472, 5849978617585;
%F Formula
The row polynomials satisfy (n+3+(2*n+6)*x+(n+3)*x^2)*P(n,x) = (6*n+48-(18*n-110)*x-(91*n-39)*x^2-(104*n+60)*x^3-(37*n+37)*x^4)*P(n-1,x) + (36*x+145*x^2+219*x^3+147*x^4+37*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 336 + 714*x + 379*x^2.
%t Mathematica
P[0, x_] := 336 + 714*x + 379*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 48 - (18*n - 110)*x - (91*n - 39)*x^2 - (104*n + 60)*x^3 - (37*n + 37)*x^4)*P[n - 1, x] + (36*x + 145*x^2 + 219*x^3 + 147*x^4 + 37*x^5)*D[P[n - 1, x], x])/(n + 3 + (2*n + 6)*x + (n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 336 + 714*x + 379*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 48 - (18*n - 110)*x - (91*n - 39)*x^2 - (104*n + 60)*x^3 - (37*n + 37)*x^4)*v[i - 1] + (36*x + 145*x^2 + 219*x^3 + 147*x^4 + 37*x^5)*deriv(v[i - 1], x))/(n + 3 + (2*n + 6)*x + (n + 3)*x^2)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741