Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.

OEIS matches ?

• A001026: 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,           120,            25;
  1:         1728,          2160,           900,           125;
  2:        20736,         34560,         21600,          6000,           625;
  3:       248832,        518400,        432000,        180000,         37500,          3125;
  4:      2985984,       7464960,       7776000,       4320000,       1350000,        225000,         15625;
  5:     35831808,     104509440,     130636800,      90720000,      37800000,       9450000,       1312500,        78125;
  6:    429981696,    1433272320,    2090188800,    1741824000,     907200000,     302400000,      63000000,      7500000,       390625;
  7:   5159780352,   19349176320,   32248627200,   31352832000,   19595520000,    8164800000,    2268000000,    405000000,     42187500,     1953125;
  8:  61917364224,  257989017600,  483729408000,  537477120000,  391910400000,  195955200000,   68040000000,  16200000000,   2531250000,   234375000,    9765625;
  9: 743008370688, 3405455032320, 7094697984000, 8868372480000, 7390310400000, 4311014400000, 1796256000000, 534600000000, 111375000000, 15468750000, 1289062500, 48828125;

The row polynomials satisfy P(n,x) = (12+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 120*x + 25*x^2.

P[0, x_] := 144 + 120*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 5*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 + 120*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 5*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-49423b91d6bf9513
• 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-49423b91d6bf9513, e-b8d4501ecbbc70e3
• 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-49423b91d6bf9513, e-b8d4501ecbbc70e3
• 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-49423b91d6bf9513
• 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-b8d4501ecbbc70e3
• 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-ce38127e6f6c296e
• 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-ce38127e6f6c296e

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-ce38127e6f6c296e root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-b8d4501ecbbc70e3 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-49423b91d6bf9513 root: theorem check

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

Used in 7 proof steps

• e-2f341f062a4ee4be basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
• e-633275fbd6b7b077 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
• e-df900b884626232f basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
• e-e0432c3ecb510b0f basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
• e-fd4b406e93b80ec2 basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
• e-d496cfbb04db8f55 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
• e-d496cfbb04db8f55 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.

Related sequences ?

Same fingerprint (2)

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

• A013620 → basis A013611
• A038271 → basis A007318

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