Real-rooted atlas

A013618 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(45 + 99\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2025 + 8910\,x + 9801\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
020258910980120736
19112560142513231359702992985984
241006253608550011908215017465382096059601429981696
3184528125202980937589311612501964855475021613410225950990049961917364224
483037656251096097062506028533843751768369927500291781038037525676731347309414801494018916100448256
5373669453125575450957812537979763215625139259131790625306370089939375404408518719975296566247061315932065347906991283918464548864
616815125390625295946206875000227878579293750010026657488925000275733080945437504852902224639700053381924471036700335543525246516409227446944279201184884258895036416
7756680642578125149822767230468751318440351628125006767993805024375002233437955658043750491356350244769625072065598035899545006794756386241957100373711601243307640591351724748364089926623333280885243904

All 15 rows

Technical details
Node id
n-436c06bbad4746ea · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-2235b7dd30e0ad6a

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:                2025,                 8910,                  9801;
  1:               91125,               601425,               1323135,                 970299;
  2:             4100625,             36085500,             119082150,              174653820,                96059601;
  3:           184528125,           2029809375,            8931161250,            19648554750,             21613410225,              9509900499;
  4:          8303765625,         109609706250,          602853384375,          1768369927500,           2917810380375,           2567673134730,            941480149401;
  5:        373669453125,        5754509578125,        37979763215625,        139259131790625,         306370089939375,         404408518719975,         296566247061315,           93206534790699;
  6:      16815125390625,      295946206875000,      2278785792937500,      10026657488925000,       27573308094543750,       48529022246397000,       53381924471036700,        33554352524651640,         9227446944279201;
  7:     756680642578125,    14982276723046875,    131844035162812500,     676799380502437500,     2233437955658043750,     4913563502447696250,     7206559803589954500,      6794756386241957100,      3737116012433076405,       913517247483640899;
  8:   34050628916015625,   749113836152343750,   7416226977908203125,   43508531603728125000,   167507846674353281250,   442220715220292662500,   810737977903869881250,   1019213457936293565000,    840851102797442191125,    411082761367638404550,    90438207500880449001;
  9: 1532278301220703125, 37081134889541015625, 407892483784951171875, 2692090392980677734375, 11845197729114982031250, 36483209005674144656250, 80263059812483118243750, 126127665419616328668750, 138740431961577961535625, 101742983438490505126125, 44766912712935822255495, 8953382542587164451099;
%F Formula
The row polynomials satisfy P(n,x) = (45+99*x)*P(n-1,x) for n >= 1, with P(0,x) = 2025 + 8910*x + 9801*x^2.
%t Mathematica
P[0, x_] := 2025 + 8910*x + 9801*x^2;
P[n_, x_] := P[n, x] = Expand[(45 + 99*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] = 2025 + 8910*x + 9801*x^2; for(n = 1, N, my(i = n + 1); v[i] = (45 + 99*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.