Real-rooted atlas

A137320 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(144\,x^{2} - 16\,n x - n^{2}\right)\,P_{n-2}(x) + \left(64\,x^{3} + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(1 + 16\,x + 2\,n\right)\,P_{n-1}(x) + \left(-16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121618
2680144230
324416129610242760
412024641022414336640033544
5720167048352015872012800036864424528
65040128448734976170496018880009953282007045657456
740320110822470358401886617625760000188375047024640104857679721280
836288010619136732965762198282243497536003179151361657815044613734453084161189002816
93628800111997440828537984271950233648714624005193732096338868633613274972162866544642621440018757913472
1039916800129003264010121894784358063513607058297600085190676480655525355523230662656098736537601703936000126877696312595477632
11479001600161190604801330730311685016541347841070785408000142884605952012504463319047301580718082820361420806920601600097695825926039797765493176819712
126227020800217211811840187502094950474678008627201705554864640024742722723840240350753054721602499903488073628950855682292449280000461961691136543581798402835349504101599105941504
138717829120031405976985602820519959961611788269438566428545143644160044463807833702447120143355084834992099216588818420708881203268454265651200175629181911042960104882176294876348416131533373441974020017692672
1413076743680004850381813760045134337328128019688107240488965018538386534400831377144895897694888288186531847710601963896832453458135502028819364786208768005947624186183681280068697456641831919314534415652471439366039797760040215480310407168
Technical details
Node id
n-9e538a84352d383f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-eb28e60e59f86df9

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       2,        16;
  2:       6,        80,       144;
  3:      24,       416,      1296,       1024;
  4:     120,      2464,     10224,      14336,       6400;
  5:     720,     16704,     83520,     158720,     128000,      36864;
  6:    5040,    128448,    734976,    1704960,    1888000,     995328,     200704;
  7:   40320,   1108224,   7035840,   18866176,   25760000,   18837504,    7024640,    1048576;
  8:  362880,  10619136,  73296576,  219828224,  349753600,  317915136,  165781504,   46137344,   5308416;
  9: 3628800, 111997440, 828537984, 2719502336, 4871462400, 5193732096, 3388686336, 1327497216, 286654464, 26214400;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+16*n*x-144*x^2)*P(n-2,x) + (16*n*x^2+64*x^3)*d/dx P(n-2,x) + (2*n+1+16*x)*P(n-1,x) - 16*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 16*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 16*n*x - 144*x^2)*P[n - 2, x] + (16*n*x^2 + 64*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 16*x)*P[n - 1, x] - 16*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 16*n*x - 144*x^2)*v[i - 2] + (16*n*x^2 + 64*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 16*x)*v[i - 1] - 16*x^2*deriv(v[i - 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

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.