Real-rooted atlas

A034870 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(36 + 144\,x + 144\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
136144144324
2129610368311044147220736104976
34665655987227993607464960111974408957952298598434012224
4167961626873856188116992752467968188116992030098718723009871872171992678442998169611019960576
56046617612093235201088391168058047528960203166351360487599243264812665405440928760463360696570347520309586821120619173642243570467226624
6217678233652242776064574670536704383113691136017240116101120551683715235841287262002216962206734860943362758418576179202451927623270401471156573962245349660268953689161004482561156831381426176
778364164096219419659468828524555730944228196445847552125508045216153650203218086461441506096542593843234425063830716416602438617037537288032514893833830480325148938338304584182901369733122920914506848665689874292518420481283918464548864374813367582081024

All 15 rows

Technical details
Node id
n-3dda967a9c099e56 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-90334921cba1a1bb

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:              36,              144,               144;
  2:            1296,            10368,             31104,              41472,               20736;
  3:           46656,           559872,           2799360,            7464960,            11197440,              8957952,               2985984;
  4:         1679616,         26873856,         188116992,          752467968,          1881169920,           3009871872,            3009871872,            1719926784,              429981696;
  5:        60466176,       1209323520,       10883911680,        58047528960,        203166351360,         487599243264,          812665405440,          928760463360,           696570347520,           309586821120,            61917364224;
  6:      2176782336,      52242776064,      574670536704,      3831136911360,      17240116101120,       55168371523584,       128726200221696,       220673486094336,        275841857617920,        245192762327040,        147115657396224,         53496602689536,          8916100448256;
  7:     78364164096,    2194196594688,    28524555730944,    228196445847552,    1255080452161536,     5020321808646144,     15060965425938432,     34425063830716416,      60243861703753728,      80325148938338304,      80325148938338304,      58418290136973312,      29209145068486656,       8987429251842048,       1283918464548864;
  8:   2821109907456,   90275517038592,  1354132755578880,  12638572385402880,   82150720505118720,   394323458424569856,   1445852680890089472,   4131007659685969920,    9294767234293432320,   16524030638743879680,   23133642894241431552,   25236701339172470784,   21030584449310392320,   12941898122652549120,    5546527766851092480,    1479074071160291328,     184884258895036416;
  9: 101559956668416, 3656158440062976, 62154693481070592, 662983397131419648, 4972375478485647360, 27845302679519625216, 120662978277918375936, 413701639810005860352, 1137679509477516115968, 2528176687727813591040, 4550718037910064463872, 6619226236960093765632, 7722430609786776059904, 7128397485957024055296, 5091712489969302896640, 2715579994650294878208, 1018342497993860579328, 239609999527967195136, 26623333280885243904;
%F Formula
The row polynomials satisfy P(n,x) = (36+144*x+144*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[(36 + 144*x + 144*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] = (36 + 144*x + 144*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

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