Real-rooted atlas

A038284 → basis A007318 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 480\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
04004801441024
18000144008640172832768
2160000384000345600138240207361048576
332000009600000115200006912000207360024883233554432
4640000002304000003456000002764800001244160002985984029859841073741824
51280000000537600000096768000009676800000580608000020901888004180377603583180834359738368
6256000000001228800000002580480000003096576000002322432000001114767360003344302080057330892804299816961099511627776
751200000000027648000000006635520000000928972800000083607552000005016453120000200658124800051597803520077396705280515978035235184372088832
810240000000000614400000000001658880000000002654208000000002786918400000002006581248000001003290624000003439853568000077396705280001031956070400619173642241125899906842624
920480000000000013516800000000004055040000000000729907200000000087588864000000007357464576000000441447874560000018919194624000005675758387200001135151677440001362182012928074300837068836028797018963968
104096000000000000294912000000000009732096000000000019464192000000000026276659200000000025225592832000000017657914982400000090812134195200000340545503232000009081213419520000163461841551360017832200896512089161004482561152921504606846976
1181920000000000000638976000000000000230031360000000000050606899200000000007591034880000000000819831767040000000065586541363200000003935192481792000000177083661680640000059027887226880000014166692934451200023181861165465600231818611654656010699320537907236893488147419103232
12163840000000000000013762560000000000000536739840000000000001288175616000000000002125489766400000000002550587719680000000002295528947712000000001574076992716800000008263904211763200000033055616847052800000991668505411584000021636403754434560003245460563165184002995809750614016012839184645488641180591620717411303424
1332768000000000000000294912000000000000000123863040000000000000032204390400000000000005796790272000000000000765176315904000000000076517631590400000000005902788722688000000000354167323361280000000016527808423526400000005950011032469504000001622730281582592000003245460563165184000044937146259210240003851755393646592001540702157458636837778931862957161709568
146553600000000000000006291456000000000000000283115520000000000000007927234560000000000000015458107392000000000000022259674644480000000000024485642108928000000000020987693236224000000000014166692934451200000000075555695650406400000000317333921731706880000001038547380212858880000025963684505321472000004793295600982425600006162808629834547200049302469038676377601848842588950364161208925819614629174706176
Technical details
Node id
n-6ff233938df68c62 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-b1029f17d1b64336

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:             400,              480,              144;
  1:            8000,            14400,             8640,             1728;
  2:          160000,           384000,           345600,           138240,            20736;
  3:         3200000,          9600000,         11520000,          6912000,          2073600,           248832;
  4:        64000000,        230400000,        345600000,        276480000,        124416000,         29859840,          2985984;
  5:      1280000000,       5376000000,       9676800000,       9676800000,       5806080000,       2090188800,        418037760,         35831808;
  6:     25600000000,     122880000000,     258048000000,     309657600000,     232243200000,     111476736000,      33443020800,       5733089280,       429981696;
  7:    512000000000,    2764800000000,    6635520000000,    9289728000000,    8360755200000,    5016453120000,    2006581248000,     515978035200,     77396705280,      5159780352;
  8:  10240000000000,   61440000000000,  165888000000000,  265420800000000,  278691840000000,  200658124800000,  100329062400000,   34398535680000,   7739670528000,   1031956070400,    61917364224;
  9: 204800000000000, 1351680000000000, 4055040000000000, 7299072000000000, 8758886400000000, 7357464576000000, 4414478745600000, 1891919462400000, 567575838720000, 113515167744000, 13621820129280, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (20+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 480*x + 144*x^2.
%t Mathematica
P[0, x_] := 400 + 480*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 12*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] = 400 + 480*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 12*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.