Real-rooted atlas

A038243 → basis A013609

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(6 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
23624464
3216216728512
412961728864192164096
5777612960864028804803232768
6466569331277760345608640115264262144
72799366531846531843628801209602419226881282097152
81679616447897652254723483648145152038707264512614425616777216
910077696302330884031078431352832156764165225472116121616588813824512134217728
1060466176201553920302330880268738560156764160627056641741824033177604147203072010241073741824
11362797056133025587222170931202217093120147806208068976230422992076854743040912384010137606758420488589934592
122176782336870712934415963070464177367449601330255872070946979842759049216788299776164229120243302402433024147456409668719476736
13130606940165659634073611319268147213834661068811528884224069173305344307436912641024789708825619742724744396806325862457507843194888192549755813888
14783641640963656994324487923487703041056465027072968426274816645617516544322808758272122974765056358676398087970586624132843110416102195213418496688128163844398046511104

Generated b-files: A001018 (row sums), A038212 (reversed triangle), A038256 ?

Computed in your browser ?

Technical details
Node id
n-ff3307aeb79ab12b · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f9762973ec1089d7

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        6,        2;
  2:       36,       24,        4;
  3:      216,      216,       72,        8;
  4:     1296,     1728,      864,      192,       16;
  5:     7776,    12960,     8640,     2880,      480,      32;
  6:    46656,    93312,    77760,    34560,     8640,    1152,      64;
  7:   279936,   653184,   653184,   362880,   120960,   24192,    2688,    128;
  8:  1679616,  4478976,  5225472,  3483648,  1451520,  387072,   64512,   6144,   256;
  9: 10077696, 30233088, 40310784, 31352832, 15676416, 5225472, 1161216, 165888, 13824, 512;
%F Formula
The row polynomials satisfy P(n,x) = (6+2*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[(6 + 2*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] = (6 + 2*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)

Generated with polytool c48629374597.

A038212: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,     6;
  2:   4,    24,     36;
  3:   8,    72,    216,     216;
  4:  16,   192,    864,    1728,    1296;
  5:  32,   480,   2880,    8640,   12960,     7776;
  6:  64,  1152,   8640,   34560,   77760,    93312,    46656;
  7: 128,  2688,  24192,  120960,  362880,   653184,   653184,   279936;
  8: 256,  6144,  64512,  387072, 1451520,  3483648,  5225472,  4478976,  1679616;
  9: 512, 13824, 165888, 1161216, 5225472, 15676416, 31352832, 40310784, 30233088, 10077696;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+2*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[(6 + 2*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (6 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A038256: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,        2;
  2:       36,       24,        4;
  3:      216,      216,       72,        8;
  4:     1296,     1728,      864,      192,       16;
  5:     7776,    12960,     8640,     2880,      480,      32;
  6:    46656,    93312,    77760,    34560,     8640,    1152,      64;
  7:   279936,   653184,   653184,   362880,   120960,   24192,    2688,    128;
  8:  1679616,  4478976,  5225472,  3483648,  1451520,  387072,   64512,   6144,   256;
  9: 10077696, 30233088, 40310784, 31352832, 15676416, 5225472, 1161216, 165888, 13824, 512;
%F Formula
The row polynomials satisfy P(n,x) = (6+2*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[(6 + 2*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (6 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-037f2552a98ec0cc
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-037f2552a98ec0cc
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-037f2552a98ec0cc

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (16)

Equal up to content, sign, reversal and a small index shift.

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. [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
  4. [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