Real-rooted atlas

A038319 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(36 + 60\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1296 + 4320\,x + 3600\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01296432036009216
146656233280388800216000884736
216796161119744027993600311040001296000084934656
3604661765038848001679616000279936000023328000007776000008153726976
42176782336217678233609069926400020155392000025194240000016796160000046656000000782757789696
578364164096914248581120457124290560012697896960000211631616000002116316160000011757312000000279936000000075144747810816
62821109907456376147987660802194196594688007313988648960001523747635200000203166351360000016930529280000008062156800000001679616000000007213895789838336
710155995666841615233993500262401015599566684160039495538704384000987388467609600001645647446016000001828497162240000001306069401600000005441955840000000010077696000000000692533995824480256
8365615844006297660935974001049600457019805007872000203119913336832000059243308056576000001184866161131520000016456474460160000000156728328192000000009795520512000000000362797056000000000060466176000000000066483263599150104576
9131621703842267136241306457044156416020108871420346368000100544357101731840000335147857005772800000782011666346803200000130335277724467200000015516104491008000000001293008707584000000000718338170880000000000239446056960000000000362797056000000000006382393305518410039296
10473838133832161689694767626766432337920868703245358963097600482612914088312832000018097984278311731200000482612914088312832000009384139996161638400000013405914280230912000000013964494041907200000000010344069660672000000000051720348303360000000000156728328192000000000002176782336000000000000612709757329767363772416
111705817281795782082563695937443890861178880369593744389086117888002258628437933304053760009410951824722100224000002823285547416630067200000627396788314806681600000010456613138580111360000000130707664232251392000000001210256150298624000000000080683743353241600000000003667442879692800000000000101873413324800000000000013060694016000000000000058820136703657666922151936
126140942214464815497216143288651670845694935040155229372643416169512960010348624842894411300864000474311971965993851289600001581039906553312837632000003952599766383282094080000007528761459777680179200000001097944379550911692800000000121993819950101299200000000010166151662508441600000000006161304037883904000000000002567210015784960000000000006582589784064000000000000078364164096000000000000005646733123551136024526585856
13221073919720733357899776552684799301833394749440064479893251880562720768000465688117930248508538880000232844058965124254269440000085376154953878893232128000002371559859829969256448000000050819139853499341209600000000846985664224989020160000000001097944379550911692800000000001097944379550911692800000000008317760451143270400000000000046209780284129280000000000000177729924169728000000000000004231664861184000000000000000470184984576000000000000000542086379860909058354552242176
14795866110994640088439193621223096293190402358378496026528870366488002947973120002063356584060178007064576000011176514830325964204933120000044706059321303856819732480000013660184792620622917140480000003252424950623957837414400000000609829678241992094515200000000090345137517332162150400000000001054026604368875225088000000000095820600397170475008000000000006654208360914616320000000000000341241454405877760000000000000012187194800209920000000000000002708265511157760000000000000002821109907456000000000000000052040292466647269602037015248896
Technical details
Node id
n-263d5a8e971d7391 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-b1029f17d1b64336

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:               1296,                4320,                 3600;
  1:              46656,              233280,               388800,                216000;
  2:            1679616,            11197440,             27993600,              31104000,              12960000;
  3:           60466176,           503884800,           1679616000,            2799360000,            2332800000,             777600000;
  4:         2176782336,         21767823360,          90699264000,          201553920000,          251942400000,          167961600000,            46656000000;
  5:        78364164096,        914248581120,        4571242905600,        12697896960000,        21163161600000,        21163161600000,         11757312000000,          2799360000000;
  6:      2821109907456,      37614798766080,      219419659468800,       731398864896000,      1523747635200000,      2031663513600000,       1693052928000000,        806215680000000,        167961600000000;
  7:    101559956668416,    1523399350026240,    10155995666841600,     39495538704384000,     98738846760960000,    164564744601600000,     182849716224000000,     130606940160000000,      54419558400000000,     10077696000000000;
  8:   3656158440062976,   60935974001049600,   457019805007872000,   2031199133368320000,   5924330805657600000,  11848661611315200000,   16456474460160000000,   15672832819200000000,    9795520512000000000,   3627970560000000000,    604661760000000000;
  9: 131621703842267136, 2413064570441564160, 20108871420346368000, 100544357101731840000, 335147857005772800000, 782011666346803200000, 1303352777244672000000, 1551610449100800000000, 1293008707584000000000, 718338170880000000000, 239446056960000000000, 36279705600000000000;
%F Formula
The row polynomials satisfy P(n,x) = (36+60*x)*P(n-1,x) for n >= 1, with P(0,x) = 1296 + 4320*x + 3600*x^2.
%t Mathematica
P[0, x_] := 1296 + 4320*x + 3600*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 60*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] = 1296 + 4320*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 60*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.