{"id":"n-be41df8499a2681b","revision":1,"first_index":1,"fingerprint":"f6-14fed6d19df1fb72","name":"Triangle read by rows. 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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-a903e57fab33046a"]},{"id":169962,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-a903e57fab33046a","e-f7a0355494cd983e"]},{"id":169963,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-a903e57fab33046a","e-f7a0355494cd983e"]},{"id":169964,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2","region_exact":null,"edges":["e-a903e57fab33046a"]},{"id":170079,"position":4,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-f7a0355494cd983e"]}],"region_exact":{},"incoming_edges":[{"id":"e-a903e57fab33046a","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-be41df8499a2681b","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; S_n = A + B D + C D^2 with A = 2 + x, B = 8 + 8 x, C = 16 x: Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5); (i) is necessary for a real-rootedness preserver (J. Borcea and P. Branden, Proc. London Math. Soc. 101 (2010), Theorem 1.3); with coefficients of one sign T also keeps every root in (-inf, 0]; a linear operator preserving real-rootedness preserves the alternation of two polynomials (P. Branden, Trans. Amer. Math. Soc. 358 (2006), Corollary 2.5)","lean_route":null,"lean_declarations":[],"certificate_hash":"78c12b479cd2fd9904f099e93f133b6007c9ee1dc784db60d5db9e6895dd1ac1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},{"id":"e-f7a0355494cd983e","operation":{"operation":"theorem_check","theorem":"hypergeometric_row"},"operation_detail":"hypergeometric_row","conclusion_id":"n-be41df8499a2681b","premise_count":0,"theorem":"Row-hypergeometric rows: an exact row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, proved for every n by an invariant row-annihilator system of the rule, writes every row as U_0(n) x^v (or its reversal) times the image of (1 + x)^M(n) under a product of nonnegative multiplier sequences: C^k (C > 0); k -> M'!/(M' - k)!, whose exponential generating function is (1 + x)^M' (Polya--Schur, Theorem 7.1 (3) => (1) of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601: a sequence is a multiplier sequence when G(x) = sum_k lambda_k x^k / k! or G(-x) can be written C x^m e^(a x) prod_k (1 + alpha_k x) with a >= 0, alpha_k >= 0); k -> phi(k) for a polynomial phi with only real roots, all <= 0 (Laguerre's theorem, RealRooted isPFMultiplierSequence_eval_of_roots_nonpos); k -> 1/(alpha)_k for alpha > 0 (RealRooted isPFMultiplierSequence_inv_ascPochhammer, via the generalized Laguerre polynomials); products of PF multiplier sequences are PF multiplier sequences (RealRooted IsPFMultiplierSequence.mul) and map real-rooted polynomials with nonnegative coefficients to such polynomials (IsPFMultiplierSequence.diagonalOperator_isPF)","lean_route":null,"lean_declarations":["RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_inv_ascPochhammer","RealRooted.IsPFMultiplierSequence.mul","RealRooted.IsPFMultiplierSequence.diagonalOperator_isPF"],"certificate_hash":"553eedb638a72ba064b7f38a34c70210c1356072e15f24d0591ebb4a3e1e82a5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-3f86cc05e6d83baf","operation":"preserver_image","operation_detail":"affine, shift 2","conclusion_id":"n-aceeb2572b6504c7","conclusion_name":"Coefficient array of orthogonal polynomials whose moment sequence is the double factorial numbers A001147."}],"used_by_total":1,"links":[{"id":"l-ffeedf6f6b5e343d","kind":"oeis_match","node_a_id":"n-be41df8499a2681b","node_b_id":null,"fingerprint":null,"oeis_id":"A048854","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A048854","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A048854","relation":"identification"},{"oeis_id":"A048854","relation":"match_triangle"}],"bfiles":[{"node_id":"n-be41df8499a2681b","oeis_id":"A048854","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":37169}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:           1;\n   2:           2,           1;\n   3:          12,          12,           1;\n   4:         120,         180,          30,           1;\n   5:        1680,        3360,         840,          56,          1;\n   6:       30240,       75600,       25200,        2520,         90,        1;\n   7:      665280,     1995840,      831600,      110880,       5940,      132,       1;\n   8:    17297280,    60540480,    30270240,     5045040,     360360,    12012,     182,     1;\n   9:   518918400,  2075673600,  1210809600,   242161920,   21621600,   960960,   21840,   240,   1;\n  10: 17643225600, 79394515200, 52929676800, 12350257920, 1323241920, 73513440, 2227680, 36720, 306, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (8+8*x)*d/dx P(n-1,x) + 16*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (8 + 8*x)*D[P[n - 1, x], x] + 16*x*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + (8 + 8*x)*deriv(v[i - 1], x) + 16*x*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A048854","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:           1;\n  1:           2,           1;\n  2:          12,          12,           1;\n  3:         120,         180,          30,           1;\n  4:        1680,        3360,         840,          56,          1;\n  5:       30240,       75600,       25200,        2520,         90,        1;\n  6:      665280,     1995840,      831600,      110880,       5940,      132,       1;\n  7:    17297280,    60540480,    30270240,     5045040,     360360,    12012,     182,     1;\n  8:   518918400,  2075673600,  1210809600,   242161920,   21621600,   960960,   21840,   240,   1;\n  9: 17643225600, 79394515200, 52929676800, 12350257920, 1323241920, 73513440, 2227680, 36720, 306, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (8+8*x)*d/dx P(n-1,x) + 16*x*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (8 + 8*x)*D[P[n - 1, x], x] + 16*x*D[P[n - 1, x], {x, 2}]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + x)*v[i - 1] + (8 + 8*x)*deriv(v[i - 1], x) + 16*x*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-be41df8499a2681b","steps":[],"edges":[],"text":"A048854"},"headline":"A048854: Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].","chain":null}