Real-rooted atlas

A038221 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(6 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
161218
236144144324
32161296259217285832
4129610368311044147220736104976
57776777603110406220806220802488321889568
64665655987227993607464960111974408957952298598434012224
72799363919104235146247838208015676416018811699212541132835831808612220032
8167961626873856188116992752467968188116992030098718723009871872171992678442998169611019960576
9100776961813985281451188224677221171220316635136406332702725417769369646438023168232190115845159780352198359290368
106046617612093235201088391168058047528960203166351360487599243264812665405440928760463360696570347520309586821120619173642243570467226624
113627970567981535232798153523204788921139201915568455680536359167590410727183351808153245476454401532454764544010216365096960408654603878474300837068864268410079232
12217678233652242776064574670536704383113691136017240116101120551683715235841287262002216962206734860943362758418576179202451927623270401471156573962245349660268953689161004482561156831381426176
131306069401633957804441640749365329922988286790860814941433954304053789162235494414343776596131842868755319226368430313297883955247812588653772803825007092301824208636750489190469545583496396810699320537907220822964865671168
1478364164096219419659468828524555730944228196445847552125508045216153650203218086461441506096542593843234425063830716416602438617037537288032514893833830480325148938338304584182901369733122920914506848665689874292518420481283918464548864374813367582081024

Generated b-files: A001027 (row sums), A038266, A038332 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-feee915c2922c082 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-11aec07ce60b6d61

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,        12;
  2:       36,       144,        144;
  3:      216,      1296,       2592,       1728;
  4:     1296,     10368,      31104,      41472,       20736;
  5:     7776,     77760,     311040,     622080,      622080,      248832;
  6:    46656,    559872,    2799360,    7464960,    11197440,     8957952,     2985984;
  7:   279936,   3919104,   23514624,   78382080,   156764160,   188116992,   125411328,    35831808;
  8:  1679616,  26873856,  188116992,  752467968,  1881169920,  3009871872,  3009871872,  1719926784,   429981696;
  9: 10077696, 181398528, 1451188224, 6772211712, 20316635136, 40633270272, 54177693696, 46438023168, 23219011584, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (6+12*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 + 12*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 + 12*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.

A038266: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,        12;
  2:       36,       144,        144;
  3:      216,      1296,       2592,       1728;
  4:     1296,     10368,      31104,      41472,       20736;
  5:     7776,     77760,     311040,     622080,      622080,      248832;
  6:    46656,    559872,    2799360,    7464960,    11197440,     8957952,     2985984;
  7:   279936,   3919104,   23514624,   78382080,   156764160,   188116992,   125411328,    35831808;
  8:  1679616,  26873856,  188116992,  752467968,  1881169920,  3009871872,  3009871872,  1719926784,   429981696;
  9: 10077696, 181398528, 1451188224, 6772211712, 20316635136, 40633270272, 54177693696, 46438023168, 23219011584, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (6+12*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 + 12*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 + 12*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.

A038332: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         12,           6;
  2:        144,         144,          36;
  3:       1728,        2592,        1296,         216;
  4:      20736,       41472,       31104,       10368,        1296;
  5:     248832,      622080,      622080,      311040,       77760,        7776;
  6:    2985984,     8957952,    11197440,     7464960,     2799360,      559872,      46656;
  7:   35831808,   125411328,   188116992,   156764160,    78382080,    23514624,    3919104,     279936;
  8:  429981696,  1719926784,  3009871872,  3009871872,  1881169920,   752467968,  188116992,   26873856,   1679616;
  9: 5159780352, 23219011584, 46438023168, 54177693696, 40633270272, 20316635136, 6772211712, 1451188224, 181398528, 10077696;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+12*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 + 12*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 + 12*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.

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-fc7ebcb9132c171a
  • 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-fc7ebcb9132c171a
  • 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-fc7ebcb9132c171a

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 (67)

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