Real-rooted atlas

A038329 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(15 + 39\,x + 24\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 225 + 720\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02257205761521
1337516200259201382459319
2506253240007776008294403317762313441
37593756075000194400003110400024883200796262490224199
41139062510935000043740000093312000011197440007166361601911029763518743761
517085937519136250009185400000244944000003919104000037623398400200658124804586471424137231006679
62562890625328050000001837080000005878656000001175731200000150493593600012039487488005503765708801100753141765352009260481
73844335937555358437500035429400000001322697600000031744742400000507915878400005417769369600037150418534400148601674137602641807540224208728361158759
85766503906259226406250000664301250000002834352000000007936185600000001523747635200000203166351360000018575209267200001114512556032000396271131033600634033809653768140406085191601
9864975585937515223570312500012178856250000005845851000000000187067232000000004190305996800000067044895948800000766227382272000006129819058176000032692368310272000104615578592870401521681143169024317475837322472439
10129746337890625249112968750000021921941250000000116917020000000000420901272000000000107750725632000000020113468784640000002758418576179200000275841857617920000019615420986163200009415402073358336002739026057704243203652034743605657612381557655576425121
11194619506835937540480857421875000388616231250000000227988189000000000091195275600000000002626423937280000000056030377328640000000896486037258240000001075783244709888000009562517730754560000061200113476829184000267055040626163712007121467750031032320876488338465357824482880748567480579719
12291929260253906256539215429687500006800784046875000000435250179000000000001915100787600000000006128322520320000000001470797404876800000000268945811177472000000037652413564846080000004016257446916915200000321300595753353216000018693852843831459840007477541137532583936001840625510777251430402103572012316858777618832349194131742609041
1343789389038085937510509453369140625000117705877734375000000816094085625000000000391725161100000000000013788725670720000000000367699351219200000000007564100939366400000000012102561502986240000000015060965425938432000000014458526808900894720000010515292224655196160000056081558531494379520000207070369962440785920004733037027712932249600504857282956046106624734461618571137961752599
1465684083557128906251681512539062500000002017815046875000000000150663523500000000000007834503222000000000000030084492372480000000000088247844292608000000000020170935838310400000000003630768450895872000000000516375957460746240000000057834107235603578880000005047340267834494156800000336489351188966277120000016565629596995262873600005679644433255518699520001211657479094510655897601211657479094510655897628644003124274380508351361
Technical details
Node id
n-bb08c7dea0976170 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4b28493176e02e0b

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:           225,             720,              576;
  1:          3375,           16200,            25920,            13824;
  2:         50625,          324000,           777600,           829440,            331776;
  3:        759375,         6075000,         19440000,         31104000,          24883200,           7962624;
  4:      11390625,       109350000,        437400000,        933120000,        1119744000,         716636160,         191102976;
  5:     170859375,      1913625000,       9185400000,      24494400000,       39191040000,       37623398400,       20065812480,        4586471424;
  6:    2562890625,     32805000000,     183708000000,     587865600000,     1175731200000,     1504935936000,     1203948748800,      550376570880,      110075314176;
  7:   38443359375,    553584375000,    3542940000000,   13226976000000,    31744742400000,    50791587840000,    54177693696000,    37150418534400,    14860167413760,     2641807540224;
  8:  576650390625,   9226406250000,   66430125000000,  283435200000000,   793618560000000,  1523747635200000,  2031663513600000,  1857520926720000,  1114512556032000,   396271131033600,    63403380965376;
  9: 8649755859375, 152235703125000, 1217885625000000, 5845851000000000, 18706723200000000, 41903059968000000, 67044895948800000, 76622738227200000, 61298190581760000, 32692368310272000, 10461557859287040, 1521681143169024;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (15+39*x+24*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 720*x + 576*x^2.
%t Mathematica
P[0, x_] := 225 + 720*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15 + 39*x + 24*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] = 225 + 720*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15 + 39*x + 24*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

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.