Real-rooted atlas

A038259 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(26 + 30\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1263056
267615609003136
317576608407020027000175616
44569762109120365040028080008100009834496
5118813766854640015818400018252000010530000024300000550731776
63089157762138647680616917600094910400008213400000379080000072900000030840979456
7803181017664872312960224558006400431842320000498279600000344962800000132678000000218700000001727094849536
82088270645761927634442240778467755520017964640512000259105392000002391742080000013798512000000454896000000065610000000096717311574016
9542950367897656383307435520260230649702400700620979968000121261323456000013991691168000001076283936000000532228320000000153527400000000196830000000005416169448144896
10141167095653376162885110369280084574961153280002602306497024000052546573497600000727567940736000006995845584000000046126454400000000199585620000000005117580000000000590490000000000303305489096114176
11367034448698777646585141565614080268760432109312000930324572686080000214690286004480000034680738508416000004001623674048000000329804148960000000019027162440000000007318139400000000001688801400000000001771470000000000016985107389382393856
129542895666168217613213240153155993608385325481810534400322512518531174400008372921154174720000015457700592322560000020808443105049600000020579778895104000000014841186703200000000076108649760000000000263453018400000000005526986400000000000531441000000000000951166013805414055936
13248115287320373657637217293098056048640257658182986541875200109009231263536947200031444970556789504000006530878500256281600000100475053850096640000001159327544424192000000010032642211363200000000643118090472000000000029682373406400000000009340607016000000000001796270580000000000001594323000000000000053265296773103187132416
146450997470329715097610420842067455693619207815631550591770214400360721456181158625280001144596928267137945600002641377526770318336000004571614950179397120000006028503231005798400000006086469608227008000000004681899698636160000000002701095979982400000000001133326984608000000000003269212455600000000000058033357200000000000004782969000000000000002982856619293778479415296
Technical details
Node id
n-d4932ac719fd68dc · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-74360a37e7e7e405

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:             1;
  1:            26,             30;
  2:           676,           1560,             900;
  3:         17576,          60840,           70200,           27000;
  4:        456976,        2109120,         3650400,         2808000,           810000;
  5:      11881376,       68546400,       158184000,       182520000,        105300000,         24300000;
  6:     308915776,     2138647680,      6169176000,      9491040000,       8213400000,       3790800000,        729000000;
  7:    8031810176,    64872312960,    224558006400,    431842320000,     498279600000,     344962800000,     132678000000,     21870000000;
  8:  208827064576,  1927634442240,   7784677555200,  17964640512000,   25910539200000,   23917420800000,   13798512000000,   4548960000000,    656100000000;
  9: 5429503678976, 56383307435520, 260230649702400, 700620979968000, 1212613234560000, 1399169116800000, 1076283936000000, 532228320000000, 153527400000000, 19683000000000;
%F Formula
The row polynomials satisfy P(n,x) = (26+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
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] = 1; 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.