Real-rooted atlas

A136572 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
151217
250240288578
37505400129601036829478
4150001440005184008294404976642004504
5375000450000021600000518400006220800029859840170382840
611250000162000000972000000311040000055987200005374771200214990848017379049680
73937500006615000000476280000001905120000004572288000006584094720005267275776001805923123202068106911920
8157500000003024000000002540160000000121927680000003657830400000070230343680000842764124160005778953994240017336861982720281262540021120
9708750000000153090000000001469664000000008230118400000002962842624000000711082229760000011377315676160000117023818383360007021429103001600187238109413376043033168623231360
10354375000000008505000000000009185400000000000587865600000000002469035520000000007110822297600000001422164459520000000195039697305600000017553572757504000009361905470668800002246857312960512007315638665949331200
1119490625000000005145525000000000061746300000000000044457336000000000002133952128000000000071700791500800000000172081899601920000000294997542174720000000353997050609664000000283197640487731200000135934867434110976000296585165310787584001368024430532524934400
12116943750000000000336798000000000000044457336000000000000355658688000000000000192055691520000000000073749385543680000000002064982795223040000000042479646073159680000000637194691097395200000006796743371705548800000048936552276279951360000213541319023767060480004270826380475341209600279076983828635086617600
137601343750000000000237161925000000000000341513172000000000000030053159136000000000000180318954816000000000000778977884805120000000000249272923137638400000000059825501553033216000000001076859027954597888000000014358120372727971840000000137837955578188529664000009022120728754158305280000360884829150166332211200066624891535415322869760061676013426128354142489600
1453209406250000000000017878360500000000000000278902423800000000000000267746326848000000000000017671257571968000000000000848220363454464000000000003053593308436070400000000008375570217424650240000000001758869745659176550400000000281419159305468248064000000033770299116656189767680000002947226104726358379724800000176833566283581502783488000065292393704707016412364800011192981777949774242119680014678891195418548285912524800
Technical details
Node id
n-32b7f037deab8913 · 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:            5,             12;
  2:           50,            240,             288;
  3:          750,           5400,           12960,           10368;
  4:        15000,         144000,          518400,          829440,           497664;
  5:       375000,        4500000,        21600000,        51840000,         62208000,         29859840;
  6:     11250000,      162000000,       972000000,      3110400000,       5598720000,       5374771200,        2149908480;
  7:    393750000,     6615000000,     47628000000,    190512000000,     457228800000,     658409472000,      526727577600,      180592312320;
  8:  15750000000,   302400000000,   2540160000000,  12192768000000,   36578304000000,   70230343680000,    84276412416000,    57789539942400,   17336861982720;
  9: 708750000000, 15309000000000, 146966400000000, 823011840000000, 2962842624000000, 7110822297600000, 11377315676160000, 11702381838336000, 7021429103001600, 1872381094133760;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+12*n*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[(5*n + 12*n*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] = (5*n + 12*n*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.