A038281 → basis A013611

OEIS matches ?

• A009967: 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:          121,           264,            144;
  1:         1331,          4356,           4752,           1728;
  2:        14641,         63888,         104544,          76032,           20736;
  3:       161051,        878460,        1916640,        2090880,         1140480,          248832;
  4:      1771561,      11595672,       31624560,       45999360,        37635840,        16422912,         2985984;
  5:     19487171,     148811124,      487018224,      885487680,       965986560,       632282112,       229920768,        35831808;
  6:    214358881,    1870768416,     7142933952,    15584583168,     21251704320,     18546941952,     10116513792,      3153199104,      429981696;
  7:   2357947691,   23150759148,   101021494464,   257145622272,    420783745536,    459036813312,    333844955136,    156083355648,    42568187904,     5159780352;
  8:  25937424601,  282953722920,  1389045548880,  4040859778560,   7714368668160,  10098809892864,   9180736266240,   5723056373760,  2341250334720,   567575838720,   61917364224;
  9: 285311670611, 3423740047332, 18674945712720, 61118004150720, 133348372692480, 203659332839424, 222173817643008, 173122455306240, 94430430167040, 34338338242560, 7492001071104, 743008370688;

The row polynomials satisfy P(n,x) = (11+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 264*x + 144*x^2.

P[0, x_] := 121 + 264*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 121 + 264*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 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)

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-8f18b93eb5dfe7b3
• 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-8f18b93eb5dfe7b3
• 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-8f18b93eb5dfe7b3

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

• basis_map A013611 e-8f18b93eb5dfe7b3

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-da334b7c58401d78 root: theorem check

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

    • parent_provers periodic_product e-c5656743efdffc45 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.
• A038326: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
• A038314 → basis A013609

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