Real-rooted atlas

A130123 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(10 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1102434
21004805761156
310007200172801382439304
410000960003456005529603317761336336
5100000120000057600001382400016588800796262445435424
6100000014400000864000002764800004976640004777574401911029761544804416
71000000016800000012096000004838400000116121600001672151040013377208320458647142452523350144
8100000000192000000016128000000774144000002322432000004459069440005350883328003669177139201100753141761785793904896
9100000000021600000000207360000000116121600000041803776000001003290624000016052649984000165112971264009906778275840264180754022460716992766464
101000000000024000000000025920000000001658880000000069672960000000200658124800000401316249600000550376570880000495338913792000264180754022400634033809653762064377754059776
111000000000002640000000000316800000000002280960000000001094860800000000367873228800000088289574912000001513535569920000018162426839040000145299414712320006974371906191360152168114316902470188843638032384
1210000000000002880000000000038016000000000030412800000000001642291200000000063063982080000000176579149824000000363248536780800000544872805171200000581197658849280000418462314371481600182601737180282880365203474360565762386420683693101056
131000000000000031200000000000044928000000000003953664000000000023721984000000000010247897088000000003279327068160000000787038496358400000014166692934451200000188889239126016000001813336695609753600011869112916718387200474764516668735488087648833846535782481138303245565435904
1410000000000000033600000000000005241600000000000050319360000000000033210777600000000001594117324800000000057388223692800000000157407699271680000000330556168470528000000528889869552844800000634667843463413760000553891936113524736000332335161668114841600122708367385150095360210357201231685877762758702310349224820736
Technical details
Node id
n-ecce39dfb22763f2 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f055f948fcf55643

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:         10,          24;
  2:        100,         480,          576;
  3:       1000,        7200,        17280,         13824;
  4:      10000,       96000,       345600,        552960,        331776;
  5:     100000,     1200000,      5760000,      13824000,      16588800,        7962624;
  6:    1000000,    14400000,     86400000,     276480000,     497664000,      477757440,      191102976;
  7:   10000000,   168000000,   1209600000,    4838400000,   11612160000,    16721510400,    13377208320,     4586471424;
  8:  100000000,  1920000000,  16128000000,   77414400000,  232243200000,   445906944000,   535088332800,   366917713920,  110075314176;
  9: 1000000000, 21600000000, 207360000000, 1161216000000, 4180377600000, 10032906240000, 16052649984000, 16511297126400, 9906778275840, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (10+24*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[(10 + 24*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] = (10 + 24*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.