Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.

OEIS matches ?

• A009964: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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.

Triangle begins:
  0:          144,           192,             64;
  1:         1728,          3456,           2304,            512;
  2:        20736,         55296,          55296,          24576,           4096;
  3:       248832,        829440,        1105920,         737280,         245760,          32768;
  4:      2985984,      11943936,       19906560,       17694720,        8847360,        2359296,         262144;
  5:     35831808,     167215104,      334430208,      371589120,      247726080,       99090432,       22020096,        2097152;
  6:    429981696,    2293235712,     5350883328,     7134511104,     5945425920,     3170893824,     1056964608,      201326592,      16777216;
  7:   5159780352,   30958682112,    82556485632,   128421199872,   128421199872,    85614133248,    38050725888,    10871635968,    1811939328,     134217728;
  8:  61917364224,  412782428160,  1238347284480,  2201506283520,  2568423997440,  2054739197952,  1141521776640,   434865438720,  108716359680,   16106127360,   1073741824;
  9: 743008370688, 5448728051712, 18162426839040, 36324853678080, 48433138237440, 45204262354944, 30136174903296, 14350559477760, 4783519825920, 1063004405760, 141733920768, 8589934592;

The row polynomials satisfy P(n,x) = (12+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 192*x + 64*x^2.

P[0, x_] := 144 + 192*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 192*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 8*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)

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-f4f0736bd14a35db
• 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-b445a5ff4f115ce0, e-f4f0736bd14a35db
• 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-b445a5ff4f115ce0, e-f4f0736bd14a35db
• 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-f4f0736bd14a35db
• 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-b445a5ff4f115ce0
• 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-060020793db4a2af
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 12] to roots in (-inf, 0]: an input root r <= 12 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-060020793db4a2af

Proof chains

3 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

• basis_map_theorem e-060020793db4a2af root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-b445a5ff4f115ce0 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-f4f0736bd14a35db root: theorem check

  Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 7 proof steps

• e-4eca5e8e987e069c basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• e-889e30cc3b49f76a basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• e-bd5ba1e8beeebdaa basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• e-cdf5d4e7a8154a31 basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• e-ec87a016e71e40fc basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• e-acb1eb270a760ac4 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• e-acb1eb270a760ac4 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.

Related sequences ?

Same fingerprint (23)

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

• A178822 → basis A013610
• A344565 → basis A013610
• A038325 → basis A013610
• A038221 → basis A013610
• A038338 → basis A013610
• A038243 → basis A013611
• A028326 → basis A013610
• A038208 → basis A013610
• A013620 → D
• A178821 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• A196347 → basis A013610
• A120910 → basis A013609
• A178820 → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A156992 → basis A013610
• A135838 → basis A013610
• A038220 → D
• A120910 → x ↦ x/(x+1)
• A356546 → basis A013610
• A085880 → basis A013610
• A085881 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.

References

1. [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