Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^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:          81,          198,           121;
  1:         729,         2673,          3267,          1331;
  2:        6561,        32076,         58806,         47916,          14641;
  3:       59049,       360855,        882090,       1078110,         658845,         161051;
  4:      531441,      3897234,      11908215,      19405980,       17788815,        8696754,        1771561;
  5:     4782969,     40920957,     150043509,     305644185,      373565115,      273947751,      111608343,       19487171;
  6:    43046721,    420901272,    1800522108,    4401276264,     6724172070,     6574746024,     4017900348,     1403076312,      214358881;
  7:   387420489,   4261625379,   20834612964,   59417229564,   108931587534,   133138606986,   108483309396,    56824590636,    17363069361,     2357947691;
  8:  3486784401,  42616253790,  234389395845,  763935808680,  1633973813010,  2396494925748,  2440874461410,  1704737719080,   781338121245,   212215292190,   25937424601;
  9: 31381059609, 421900912521, 2578283354295, 9453705632415, 23109058212570, 39542166274842, 48329314335918, 42192258547230, 25784158001085, 10504656963405, 2567805035499, 285311670611;

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

P[0, x_] := 81 + 198*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(9 + 11*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 81 + 198*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (9 + 11*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-c85fc2223c79991e
• 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-38b35fa9e37eb25d, e-c85fc2223c79991e
• 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-38b35fa9e37eb25d, e-c85fc2223c79991e
• 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-c85fc2223c79991e
• 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-38b35fa9e37eb25d
• 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-5e6f1e18f6b2db86
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 9] to roots in (-inf, 0]: an input root r <= 9 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-5e6f1e18f6b2db86

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-5e6f1e18f6b2db86 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-38b35fa9e37eb25d root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-c85fc2223c79991e root: theorem check

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

Used in 7 proof steps

• e-21c2803b3d0372c9 basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
• e-337d052427983e0b basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
• e-73a1a035b42069f4 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
• e-9ac6ce8d07a161c9 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
• e-f8d6f67754ace198 basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
• e-0ccb5845893f0f92 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
• e-0ccb5845893f0f92 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.

Related sequences ?

Same fingerprint (3)

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

• A038277 → basis A013609
• A038281 → basis A013610
• 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