Real-rooted atlas

A038319 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(26 + 30\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 676 + 1560\,x + 900\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
067615609003136
117576608407020027000175616
24569762109120365040028080008100009834496
3118813766854640015818400018252000010530000024300000550731776
43089157762138647680616917600094910400008213400000379080000072900000030840979456
5803181017664872312960224558006400431842320000498279600000344962800000132678000000218700000001727094849536
62088270645761927634442240778467755520017964640512000259105392000002391742080000013798512000000454896000000065610000000096717311574016
7542950367897656383307435520260230649702400700620979968000121261323456000013991691168000001076283936000000532228320000000153527400000000196830000000005416169448144896
8141167095653376162885110369280084574961153280002602306497024000052546573497600000727567940736000006995845584000000046126454400000000199585620000000005117580000000000590490000000000303305489096114176
9367034448698777646585141565614080268760432109312000930324572686080000214690286004480000034680738508416000004001623674048000000329804148960000000019027162440000000007318139400000000001688801400000000001771470000000000016985107389382393856
109542895666168217613213240153155993608385325481810534400322512518531174400008372921154174720000015457700592322560000020808443105049600000020579778895104000000014841186703200000000076108649760000000000263453018400000000005526986400000000000531441000000000000951166013805414055936
11248115287320373657637217293098056048640257658182986541875200109009231263536947200031444970556789504000006530878500256281600000100475053850096640000001159327544424192000000010032642211363200000000643118090472000000000029682373406400000000009340607016000000000001796270580000000000001594323000000000000053265296773103187132416
126450997470329715097610420842067455693619207815631550591770214400360721456181158625280001144596928267137945600002641377526770318336000004571614950179397120000006028503231005798400000006086469608227008000000004681899698636160000000002701095979982400000000001133326984608000000000003269212455600000000000058033357200000000000004782969000000000000002982856619293778479415296
1316772593422857259253762902948861648371793920023446894651775310643200011723447325887655321600004058116382038034534400000103013723544042415104000001981033145077738752000000029388953251153267200000000339103306744076160000000003043234804113504000000000021068548643862720000000000110499380999280000000000004249976192280000000000000113165046540000000000000018653579100000000000000014348907000000000000000167039970680451594847256576
144360874289942887405977680508448429714844418048069670772679560923054080003751503144284049702912000014068136791065186385920000038957917267565131530240000082410978835233932083200000013584227280533065728000000001763337195069196032000000000180855096930173952000000000014607527059744819200000000009193548499140096000000000004419975239971200000000000001569221978688000000000000003879944452800000000000000059691453120000000000000004304672100000000000000009354238358105289311446368256
Technical details
Node id
n-2f533abae2b58c81 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-74360a37e7e7e405

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:              676,              1560,                900;
  1:            17576,             60840,              70200,              27000;
  2:           456976,           2109120,            3650400,            2808000,              810000;
  3:         11881376,          68546400,          158184000,          182520000,           105300000,            24300000;
  4:        308915776,        2138647680,         6169176000,         9491040000,          8213400000,          3790800000,           729000000;
  5:       8031810176,       64872312960,       224558006400,       431842320000,        498279600000,        344962800000,        132678000000,         21870000000;
  6:     208827064576,     1927634442240,      7784677555200,     17964640512000,      25910539200000,      23917420800000,      13798512000000,       4548960000000,        656100000000;
  7:    5429503678976,    56383307435520,    260230649702400,    700620979968000,    1212613234560000,    1399169116800000,    1076283936000000,     532228320000000,     153527400000000,     19683000000000;
  8:  141167095653376,  1628851103692800,   8457496115328000,  26023064970240000,   52546573497600000,   72756794073600000,   69958455840000000,   46126454400000000,   19958562000000000,   5117580000000000,    590490000000000;
  9: 3670344486987776, 46585141565614080, 268760432109312000, 930324572686080000, 2146902860044800000, 3468073850841600000, 4001623674048000000, 3298041489600000000, 1902716244000000000, 731813940000000000, 168880140000000000, 17714700000000000;
%F Formula
The row polynomials satisfy P(n,x) = (26+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 676 + 1560*x + 900*x^2.
%t Mathematica
P[0, x_] := 676 + 1560*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[(26 + 30*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] = 676 + 1560*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 30*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.