Real-rooted atlas

A038324 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(21\,n + 20\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 20\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
12020
28408001640
3264605040024000100860
4740880211680020160006400005513680
519448100740880001058400006720000016000000282576100
6490092120233377200044452800004233600000201600000038400000013902744120
71200725694068612896800163364040000207446400000148176000000564480000008960000000665014593740
828817416656019211611104005489031744000871274880000082978560000004741632000000150528000000020480000000031160683820960
96808114684980518713499808001729044999360003293419046400003920736960000002987228160000001422489600000003870720000000046080000000001437286541241780
10158856009316200136162293699600051871349980800001152696666240000016467095232000000156829478400000009957427200000000406425600000000096768000000000010240000000000065476386878792200
113669573815204220349483220495640001497785230695600003803898998592000006339831664320000007245521902080000005750414208000000003129477120000000001117670400000000002365440000000000022528000000000002952985048233528220
1284066600130133040880697715649012800419379864594768000011982281845564800000228233939915520000003043119198873600000028982087608320000000197157058560000000009388431360000000000298045440000000000056770560000000000049152000000000000132078967611899625840
13191251515296052666021857316033834590400114490703034371664000363462549315465600000778848319961712000000118681648756070400000013186849861785600000001076477539737600000000640760440320000000000271221350400000000000774918144000000000001341849600000000000010649600000000000005866507478095208381060
14432522657669534490805355042428289474648003060024244736842656000106857989498746886400002544237845208259200000043615505917855872000000553847694194995200000005274739944714240000000037676713890816000000000199347692544000000000007594197811200000000000197251891200000000000031309824000000000000022937600000000000000259028868648203816209880
Technical details
Node id
n-c5989f7729c933c8 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-640a8dc8403e6c62

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:             0;
  1:            20;
  2:           840,            800;
  3:         26460,          50400,           24000;
  4:        740880,        2116800,         2016000,          640000;
  5:      19448100,       74088000,       105840000,        67200000,        16000000;
  6:     490092120,     2333772000,      4445280000,      4233600000,      2016000000,       384000000;
  7:   12007256940,    68612896800,    163364040000,    207446400000,    148176000000,     56448000000,      8960000000;
  8:  288174166560,  1921161110400,   5489031744000,   8712748800000,   8297856000000,   4741632000000,   1505280000000,   204800000000;
  9: 6808114684980, 51871349980800, 172904499936000, 329341904640000, 392073696000000, 298722816000000, 142248960000000, 38707200000000, 4608000000000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (21*n+20*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 20.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 20;
P[n_, x_] := P[n, x] = Expand[Together[((21*n + 20*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 20; for(n = 2, N, my(i = n + 1); v[i] = ((21*n + 20*n*x)*v[i - 1])/(n - 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.