A038281 → basis A013610

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:          121,           198,             81;
  1:         1331,          3267,           2673,            729;
  2:        14641,         47916,          58806,          32076,           6561;
  3:       161051,        658845,        1078110,         882090,         360855,          59049;
  4:      1771561,       8696754,       17788815,       19405980,       11908215,        3897234,         531441;
  5:     19487171,     111608343,      273947751,      373565115,      305644185,      150043509,       40920957,        4782969;
  6:    214358881,    1403076312,     4017900348,     6574746024,     6724172070,     4401276264,     1800522108,      420901272,      43046721;
  7:   2357947691,   17363069361,    56824590636,   108483309396,   133138606986,   108931587534,    59417229564,    20834612964,    4261625379,     387420489;
  8:  25937424601,  212215292190,   781338121245,  1704737719080,  2440874461410,  2396494925748,  1633973813010,   763935808680,  234389395845,   42616253790,   3486784401;
  9: 285311670611, 2567805035499, 10504656963405, 25784158001085, 42192258547230, 48329314335918, 39542166274842, 23109058212570, 9453705632415, 2578283354295, 421900912521, 31381059609;

The row polynomials satisfy P(n,x) = (11+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 198*x + 81*x^2.

P[0, x_] := 121 + 198*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 9*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 + 198*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 9*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-f67f09ded1c0d580
• 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-f67f09ded1c0d580
• 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-f67f09ded1c0d580

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 A013610 e-f67f09ded1c0d580

  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)*9^(i-j)*11^j.
• A038277 → basis A013609
• A038323: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.

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