Real-rooted atlas

A038259 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 15\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1161531
2256480225961
340961152010800337529791
46553624576034560021600050625923521
51048576491520092160008640000405000075937528629151
616777216943718402211840002764800001944000007290000011390625887503681
726843545617616076804954521600774144000072576000004082400000127575000017085937527512614111
842949672963221225472010569646080019818086400023224320000017418240000081648000000218700000002562890625852891037441
9687194767365798205849602174327193600475634073600066886041600006270566400000391910400000015746400000003690562500003844335937526439622160671
101099511627776103079215104004348654387200010871635968000017836277760000020065812480000015676416000000083980800000000295245000000006150937500000576650390625819628286980801
1117592186044416181419418583040850403524608000239175991296000044845498368000005885971660800000551809843200000036951552000000001732104000000000541282500000000101490468750000864975585937525408476896404831
1228147497671065631665934879948801632774767247360051024211476480000107629196083200000161443794124800000176579149824000000141893959680000000831409920000000003464208000000000097430850000000001660753125000000129746337890625787662783788549761
13450359962737049654887620458577920308742865079500800106130359871078400024874303094784000004197538647244800000524692330905600000049189906022400000003458665267200000000180138816000000000067552056000000000017271832500000000026987238281250000194619506835937524417546297445042591
147205759403792793694575592174780416057632001481506816002161200055556505600055718438932316160000104472072998092800000146913852653568000000157407699271680000000129123503308800000000807021895680000000003782915136000000000012896301600000000000302257068750000000043594769531250000029192926025390625756943935220796320321
Technical details
Node id
n-e1ce1a92fab8fc7c · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-d935ec63feee39a0

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:          16,           15;
  2:         256,          480,           225;
  3:        4096,        11520,         10800,          3375;
  4:       65536,       245760,        345600,        216000,         50625;
  5:     1048576,      4915200,       9216000,       8640000,       4050000,        759375;
  6:    16777216,     94371840,     221184000,     276480000,     194400000,      72900000,      11390625;
  7:   268435456,   1761607680,    4954521600,    7741440000,    7257600000,    4082400000,    1275750000,     170859375;
  8:  4294967296,  32212254720,  105696460800,  198180864000,  232243200000,  174182400000,   81648000000,   21870000000,   2562890625;
  9: 68719476736, 579820584960, 2174327193600, 4756340736000, 6688604160000, 6270566400000, 3919104000000, 1574640000000, 369056250000, 38443359375;
%F Formula
The row polynomials satisfy P(n,x) = (16+15*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[(16 + 15*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] = (16 + 15*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.