Real-rooted atlas

A038320 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 17\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 612\,x + 289\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
03246122891225
158321652415606491342875
2104976396576561816353736835211500625
31889568892296016854480159181207516890141985752521875
434012224192735936455070960573052320405912060153344556241375691838265625
561222003240474546561146778819218051148080170483065209660707028304133369441033867364339296875
611019960576832619243522752269166085198730647046137390347204637139373442189760259685908876891269757574412251875390625
719835929036816860539681286369537212928140365727470081988514472492818780414462432118247054022724786190281872113007270544211858787649778815638671875
83570467226624337210793625601433145872908803609404420659205965543417478406760949206475525321117431022402871714169123201017065434897802134581776946020159939004492758547353515625
9642684100792326676773713786883152920920399360893327594113152016873965666581760223111323813692162107162502684870414214985137159840671263187032548021132359591765403991667922889023427189630763396549157373046875
10115683138142617613110755656163328681030918806261762143986225871564804555970729977075206884577991965358087585785009665533446140873579253050883624821209975759201521529890607108804311001356720141674027296024487285826222372297613379220508056640625
1120822964865671168255659735295184896144873850000604774450169277685394616321184552389794039552020137390626498672384253581956037390689282394940695908689843216964163262686553056890094986005158648033625810582417104488661193634865011761363336035117640749904578032905937118272717781982421875
12374813367582081024495586563802973798430423508500127002624114933254333813121024298507202228097967104563846937541962826752798783161517780671232862178650527128343552712494857032835228352448607872946599958592211842606669227758224727540265328660987841717803404248227332424959536642922961241683778265594009294139545122369384765625
136746640616477458432955774087334306611206318728688487915929602585998222510795223040732699496378058646528015223867313632996322304239634948455334201369602909852945529058159488027481944485552215950720201873542825969981366401143950076013829894409649108967909684616679201546023063823404599160336953744679459976740454620131710382508302862423051509815793144884079282928466796875
1412143953109659425177618350862476818686935041299852758774656991232057289806775623771095040175847879130734075166720398588526029663903711232690148651551362499944448931152942569298611036160989350001479879774225920830565433341133637621760549096036486638349316608282867655159783392072192111313660595285131139520323475594892281577670406546529896629508119520824377838834826948384486611918756668684815070942774902496337890625
Technical details
Node id
n-bd0141a5841b35b4 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a7a3fcc1bd498552

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:            324,             612,              289;
  1:           5832,           16524,            15606,             4913;
  2:         104976,          396576,           561816,           353736,             83521;
  3:        1889568,         8922960,         16854480,         15918120,           7516890,           1419857;
  4:       34012224,       192735936,        455070960,        573052320,         405912060,         153344556,          24137569;
  5:      612220032,      4047454656,      11467788192,      18051148080,       17048306520,        9660707028,        3041333694,         410338673;
  6:    11019960576,     83261924352,     275226916608,     519873064704,      613739034720,      463713937344,      218976025968,       59088768912,       6975757441;
  7:   198359290368,   1686053968128,    6369537212928,   14036572747008,    19885144724928,    18780414462432,    11824705402272,     4786190281872,    1130072705442,     118587876497;
  8:  3570467226624,  33721079362560,  143314587290880,  360940442065920,   596554341747840,   676094920647552,   532111743102240,   287171416912320,  101706543489780,   21345817769460,   2015993900449;
  9: 64268410079232, 667677371378688, 3152920920399360, 8933275941131520, 16873965666581760, 22311132381369216, 21071625026848704, 14214985137159840, 6712631870325480, 2113235959176540, 399166792288902, 34271896307633;
%F Formula
The row polynomials satisfy P(n,x) = (18+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 612*x + 289*x^2.
%t Mathematica
P[0, x_] := 324 + 612*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 17*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] = 324 + 612*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 17*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.