Real-rooted atlas

A038338 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(60 + 72\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3600 + 8640\,x + 5184\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
036008640518417424
12160007776009331203732482299968
212960000622080001119744008957952026873856303595776
3777600000466560000011197440000134369280008062156800193491763240074642432
4466560000003359232000001007769600000161243136000014511882240006965703475201393140695045289852801024
5279936000000023514624000000846526464000001693052928000002031663513600001462797729792005851190919168010030613004288698260569735168
61679616000000001612431360000000677221171200000016253308108800000243799621632000002340476367667200014042858206003200481469424205824072220413630873692170395205042176
710077696000000000108839116800000000522427760640000000146279772979200000026330359136256000003159643096350720000252771447708057600012999674453557248003899902336067174405199869781422899212166492167065567232

All 15 rows

Technical details
Node id
n-7a15286cc13dd028 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-da2debf48cb6165b

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:                 3600,                  8640,                   5184;
  1:               216000,                777600,                 933120,                  373248;
  2:             12960000,              62208000,              111974400,                89579520,                26873856;
  3:            777600000,            4665600000,            11197440000,             13436928000,              8062156800,              1934917632;
  4:          46656000000,          335923200000,          1007769600000,           1612431360000,           1451188224000,            696570347520,            139314069504;
  5:        2799360000000,        23514624000000,         84652646400000,         169305292800000,         203166351360000,         146279772979200,          58511909191680,          10030613004288;
  6:      167961600000000,      1612431360000000,       6772211712000000,       16253308108800000,       24379962163200000,       23404763676672000,       14042858206003200,        4814694242058240,         722204136308736;
  7:    10077696000000000,    108839116800000000,     522427760640000000,     1462797729792000000,     2633035913625600000,     3159643096350720000,     2527714477080576000,     1299967445355724800,      389990233606717440,       51998697814228992;
  8:   604661760000000000,   7255941120000000000,   39182082048000000000,   125382662553600000000,   263303591362560000000,   379157171562086400000,   379157171562086400000,   259993489071144960000,   116997070082015232000,    31199218688537395200,    3743906242624487424;
  9: 36279705600000000000, 478892113920000000000, 2873352683520000000000, 10344069660672000000000, 24825767185612800000000, 41707288871829504000000, 50048746646195404800000, 42898925696738918400000, 25739355418043351040000, 10295742167217340416000, 2470978120132161699840, 269561249468963094528;
%F Formula
The row polynomials satisfy P(n,x) = (60+72*x)*P(n-1,x) for n >= 1, with P(0,x) = 3600 + 8640*x + 5184*x^2.
%t Mathematica
P[0, x_] := 3600 + 8640*x + 5184*x^2;
P[n_, x_] := P[n, x] = Expand[(60 + 72*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] = 3600 + 8640*x + 5184*x^2; for(n = 1, N, my(i = n + 1); v[i] = (60 + 72*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.