Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(25 + 40\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
125401681
26252000240012802566561
3156257500015000016000096000307204096531441
4390625250000070000001120000011200000716800028672006553606553643046721
5976562578125000281250000600000000840000000806400000537600000245760000737280001310720010485763486784401
62441406252343750000103125000002750000000049500000000633600000005913600000040550400000202752000007208960000173015040025165824016777216282429536481
7610351562568359375000355468750000113750000000025025000000004004000000000480480000000043929600000003075072000000164003840000065601536000019084083200038168166400469762048026843545622876792454961
815258789062519531250000001171875000000043750000000000113750000000000218400000000000320320000000000366080000000000329472000000000234291200000000131203072000000572522496000001908408320000046976204800008053063680008589934592042949672961853020188851841
938146972656255493164062500037353515625000015937500000000004781250000000000107100000000000001856400000000000025459200000000000280051200000000002489344000000000017923276800000000104280883200000004866441216000000179683983360000051338280960000010952166604800016428249907200154618822656068719476736150094635296999121
109536743164062515258789062500001159667968750000055664062500000000189257812500000000484500000000000000969000000000000000155040000000000000020155200000000000002149888000000000000189190144000000000013759283200000000008255569920000000004064280576000000001625712230400000005202279137280000013005697843200000244813135872000032641751449600027487790694400109951162777612157665459056928801
11238418579101562541961669921875000352478027343750000187988281250000000071435546875000000002057343750000000000046633125000000000000852720000000000000001279080000000000000001591744000000000000001655413760000000000001444724736000000000001059464806400000000006519783424000000000033530314752000000000143062676275200000005007193669632000000141379585966080000031417685770240000052913997086720000634967965040640048378511622144017592186044416984770902183611232881
1259604644775390625114440917968750000010528564453125000000617675781250000000002594238281250000000008301562500000000000002103062500000000000000432630000000000000000073547100000000000000001046003200000000000000012552038400000000000000127802572800000000000001107622297600000000000081793646592000000000005141314928640000000000274203462860800000000012339155828736000000004645329253171200000001445213545431040000003651065798983680000073021315979673600001112705767309312000121386083706470400844424930131968028147497671065679766443076872509863361
131490116119384765625309944152832031250003099441528320312500001983642578125000000000912475585937500000000032119140625000000000000899335937500000000000002055625000000000000000003905687500000000000000006249100000000000000000008498776000000000000000009889484800000000000000009889484800000000000000008520171520000000000000006329270272000000000000004050732974080000000000002227903135744000000000001048425005056000000000004193700020224000000000014126147436544000000000395532128223232000000090407343593881600000016437698835251200000022869841857740800000228698418577408000014636698788954112045035996273704966461081889226673298932241
14372529029846191406258344650268554687500009012222290039062500000624847412109375000000003124237060546875000000001199707031250000000000000367910156250000000000000092503125000000000000000001942565625000000000000000034534500000000000000000000524924400000000000000000006871737600000000000000000077879692800000000000000000766815436800000000000000006572703744000000000000000049076187955200000000000000318995221708800000000000001801384781414400000000000088067700424704000000000003708113702092800000000000133492093275340800000000040683304617246720000000010355750266208256000000021612000555565056000000360200009259417600000046105601185205452800042559016478651187200252201579132747776072057594037927936523347633027360537213511521
Technical details
Node id
n-47a6042357a612c5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-deb86af49eb59db8

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:             1;
  1:            25,             40,              16;
  2:           625,           2000,            2400,             1280,              256;
  3:         15625,          75000,          150000,           160000,            96000,             30720,              4096;
  4:        390625,        2500000,         7000000,         11200000,         11200000,           7168000,           2867200,            655360,             65536;
  5:       9765625,       78125000,       281250000,        600000000,        840000000,         806400000,         537600000,         245760000,          73728000,          13107200,           1048576;
  6:     244140625,     2343750000,     10312500000,      27500000000,      49500000000,       63360000000,       59136000000,       40550400000,       20275200000,        7208960000,        1730150400,         251658240,         16777216;
  7:    6103515625,    68359375000,    355468750000,    1137500000000,    2502500000000,     4004000000000,     4804800000000,     4392960000000,     3075072000000,     1640038400000,      656015360000,      190840832000,      38168166400,       4697620480,       268435456;
  8:  152587890625,  1953125000000,  11718750000000,   43750000000000,  113750000000000,   218400000000000,   320320000000000,   366080000000000,   329472000000000,   234291200000000,   131203072000000,    57252249600000,   19084083200000,    4697620480000,    805306368000,     85899345920,     4294967296;
  9: 3814697265625, 54931640625000, 373535156250000, 1593750000000000, 4781250000000000, 10710000000000000, 18564000000000000, 25459200000000000, 28005120000000000, 24893440000000000, 17923276800000000, 10428088320000000, 4866441216000000, 1796839833600000, 513382809600000, 109521666048000, 16428249907200, 1546188226560, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (25+40*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(25 + 40*x + 16*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (25 + 40*x + 16*x^2)*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.