Real-rooted atlas

A225471 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 + 32\,x + 36\,x^{2} + 32\,n - 32\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(-32\,x^{2} + 16\,x^{3} + 32\,n x^{2}\right)\,P_{n-2}'(x) + \left(-4 + 8\,x + 8\,n\right)\,P_{n-1}(x) + \left(-8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
2329636164
338414088641282784
461442560020160512040057424
51228805611525184001740802400011521401664
6294912014450688149667846021120112000096768313639607616
78257536042821222448409804822180659250176000593510435123281921273162752
826424115201436339404817418092544881904844822987776003344302082739609611796482073645904750848
9951268147205382214778886916866048003794167398401103155200001865996697618966528001140326403732480512001835441593344
1038050725888002228987363328030089460842496176359997440005598298112000106410737664012666268876895158272004379443201126400012390480619440145408
1116742319390720010111950205747201424240708419584882968738398208301437616128000629438231347208469895053312749570359296433564876801576960000327106562949123859473532428288
128036313307545600498767465388441607291393159672627247446466406383616172282828816384003889443844915200577902036647936581048173854723978461380608182747136000538635468892012544692224200031358117662720
1341788829199237120026578813264802611204015969802454564864272646578769808588810441409173651456002518685346347089924063883637240627245310721507983363539578107985921932621250560072233851289617605066752251969536160563211159758593368801280
142340174435157278720015218446061883359232023685477490661680742416696108785315348480066992096959004672000171117637815553228802961416958915903488359898633417523200312909137864294401956237803520000872203909857282705168793600554332979206743654403686400666860581196331171840
Technical details
Node id
n-a7c4e8f8b2894c31 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-a7258691ecdb1bdc

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:           4,            8;
  2:          32,           96,           36;
  3:         384,         1408,          864,          128;
  4:        6144,        25600,        20160,         5120,          400;
  5:      122880,       561152,       518400,       174080,        24000,        1152;
  6:     2949120,     14450688,     14966784,      6021120,      1120000,       96768,       3136;
  7:    82575360,    428212224,    484098048,    221806592,     50176000,     5935104,     351232,      8192;
  8:  2642411520,  14363394048,  17418092544,   8819048448,   2298777600,   334430208,   27396096,   1179648,   20736;
  9: 95126814720, 538221477888, 691686604800, 379416739840, 110315520000, 18659966976, 1896652800, 114032640, 3732480, 51200;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(32*n-32)*x-36*x^2)*P(n-2,x) + ((32*n-32)*x^2+16*x^3)*d/dx P(n-2,x) + (8*n-4+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (32*n - 32)*x - 36*x^2)*P[n - 2, x] + ((32*n - 32)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (8*n - 4 + 8*x)*P[n - 1, x] - 8*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] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (32*n - 32)*x - 36*x^2)*v[i - 2] + ((32*n - 32)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (8*n - 4 + 8*x)*v[i - 1] - 8*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

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.