Real-rooted atlas

A038301 → reverse → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 31\,x + 20\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 440\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0121440400961
11331726013200800029791
214641106480290400352000160000923521
31610511464100532400096800008800000320000028629151
41771561193261208784600021296000029040000021120000064000000887503681
51948717124801854013528284004099480000745360000081312000004928000000128000000027512614111
62143588813117947360198414832007215084800016397920000023851520000021683200000011264000000025600000000852891037441
723579476913858459858028061526240011904889920003246788160000590325120000071554560000005575680000000253440000000051200000000026439622160671
825937424601471589538200385845985800018707684160000595244496000001298715264000001967750400000002044416000000001393920000000005632000000000010240000000000819628286980801
9285311670611570623341222051874849202000282953722920000102892262880000026190757824000004761955968000000618435840000000056221440000000003407360000000000123904000000000020480000000000025408476896404831
103138428376721684748009466406847480094664004149987936160000169772233752000004938828618240000010476303129600000016326706176000000018553075200000000014992384000000000081776640000000000270336000000000004096000000000000787662783788549761
11345227121439318159913779474608901724123063200593448274870880002697492158504000008828156155104000002140159067904000000389119830528000000053061795072000000005359777280000000000389801984000000000019329024000000000005857280000000000008192000000000000024417546297445042591
12379749833583241966635940030068011423879291264440083082758481923200041541379240961600001510595608762240000041198062057152000000856063627161600000001361919406848000000001650811402240000000001500737638400000000009922232320000000000045101056000000000000126156800000000000001638400000000000000756943935220796320321
1341772481694156511139249500749723001449953910045102000114238792912644400006231206886144240000024924827544576960000075529780438112000000017656312310208000000003210238601856000000000453973135616000000000049524342067200000000004092920832000000000000248055808000000000000010407936000000000000002703360000000000000003276800000000000000023465261991844685929951
1445949729863572161133671941421300832018227992011995568000154661750404810880000913910343301155200000398797240713231360000013293241357107712000000345278996288512000000007062524924083200000000011414181695488000000000014527140339712000000000014407081328640000000000010914455552000000000000061059891200000000000000237895680000000000000005767168000000000000000655360000000000000000727423121747185263828481
Technical details
Node id
n-f80b20ce4b64b70c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-590b4d5bb5323b01

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,           440,            400;
  1:         1331,          7260,          13200,            8000;
  2:        14641,        106480,         290400,          352000,           160000;
  3:       161051,       1464100,        5324000,         9680000,          8800000,          3200000;
  4:      1771561,      19326120,       87846000,       212960000,        290400000,        211200000,         64000000;
  5:     19487171,     248018540,     1352828400,      4099480000,       7453600000,       8131200000,       4928000000,       1280000000;
  6:    214358881,    3117947360,    19841483200,     72150848000,     163979200000,     238515200000,     216832000000,     112640000000,      25600000000;
  7:   2357947691,   38584598580,   280615262400,   1190488992000,    3246788160000,    5903251200000,    7155456000000,    5575680000000,    2534400000000,     512000000000;
  8:  25937424601,  471589538200,  3858459858000,  18707684160000,   59524449600000,  129871526400000,  196775040000000,  204441600000000,  139392000000000,   56320000000000,   10240000000000;
  9: 285311670611, 5706233412220, 51874849202000, 282953722920000, 1028922628800000, 2619075782400000, 4761955968000000, 6184358400000000, 5622144000000000, 3407360000000000, 1239040000000000, 204800000000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+31*x+20*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 440*x + 400*x^2.
%t Mathematica
P[0, x_] := 121 + 440*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((11 + 31*x + 20*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 440*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 31*x + 20*x^2)*v[i - 1])/(1 + 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

Candidate equalities ?

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.