Real-rooted atlas

A013611: Triangle of coefficients in expansion of (1+4x)^n.

A013611 nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0) · all roots real, in [−1, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A013611
\(P_{n}(x) = \left(1 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
2181625
31124864125
411696256256625
5120160640128010243125
6124240128038406144409615625
71283362240896021504286721638478125

Show all 15 stored rows

Generated b-files: A000351 (row sums), A013611, A038231 (reversed triangle) ?

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 5\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-2e0409c0014ec240 · revision 2 · JSON
Definition source
oeis_catalog:A013611
derived:mobius_x_over_x_plus_one (equal by e-6c52688ac73883a7)
Fingerprint
f6-03d7b8417365c509

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: 1,  4;
  2: 1,  8,  16;
  3: 1, 12,  48,   64;
  4: 1, 16,  96,  256,   256;
  5: 1, 20, 160,  640,  1280,   1024;
  6: 1, 24, 240, 1280,  3840,   6144,   4096;
  7: 1, 28, 336, 2240,  8960,  21504,  28672,  16384;
  8: 1, 32, 448, 3584, 17920,  57344, 114688, 131072,  65536;
  9: 1, 36, 576, 5376, 32256, 129024, 344064, 589824, 589824, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (1+4*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[(1 + 4*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] = (1 + 4*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.

A013611: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  4;
  2: 1,  8,  16;
  3: 1, 12,  48,   64;
  4: 1, 16,  96,  256,   256;
  5: 1, 20, 160,  640,  1280,   1024;
  6: 1, 24, 240, 1280,  3840,   6144,   4096;
  7: 1, 28, 336, 2240,  8960,  21504,  28672,  16384;
  8: 1, 32, 448, 3584, 17920,  57344, 114688, 131072,  65536;
  9: 1, 36, 576, 5376, 32256, 129024, 344064, 589824, 589824, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (1+4*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[(1 + 4*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] = (1 + 4*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.

A038231: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,      1;
  2:     16,      8,      1;
  3:     64,     48,     12,      1;
  4:    256,    256,     96,     16,      1;
  5:   1024,   1280,    640,    160,     20,     1;
  6:   4096,   6144,   3840,   1280,    240,    24,    1;
  7:  16384,  28672,  21504,   8960,   2240,   336,   28,   1;
  8:  65536, 131072, 114688,  57344,  17920,  3584,  448,  32,  1;
  9: 262144, 589824, 589824, 344064, 129024, 32256, 5376, 576, 36, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+4*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[(1 + 4*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] = (1 + 4*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-f45969f98fa0cee8
  • 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-36f1e280ef03dec1, e-8a33d9f811421624, e-c19dc1e19e56bc04, e-db5094ba61234dec, e-f45969f98fa0cee8
  • 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-36f1e280ef03dec1, e-8a33d9f811421624, e-c19dc1e19e56bc04, e-db5094ba61234dec, e-f45969f98fa0cee8
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-f45969f98fa0cee8
  • 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-36f1e280ef03dec1, e-8a33d9f811421624, e-c19dc1e19e56bc04
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-415345b4c6061d69
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 1] to roots in (-inf, 0]: an input root r <= 1 gives the root (r - a)/d <= 0; a larger root gives a positive root
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-415345b4c6061d69
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-db5094ba61234dec

Proof chains

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

Used in 18 proof steps

Same fingerprint (14)

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