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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":"e7933992fca0af402bc396926b03d2ad67eb3284a780cb18fec7e5eb3e9fbac4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[]},{"id":"e-dd9803bc90fda5fc","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-66d7d2fd3a2f1f0d","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 = 6 + x, B = 24 + 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":"58041b08e0b0a37f3b2234d2713a188a3f3bc22d6ce22021657530d471b317c4","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":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-ae13b7df70050976","kind":"oeis_match","node_a_id":"n-66d7d2fd3a2f1f0d","node_b_id":null,"fingerprint":null,"oeis_id":"A292219","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A292219","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A292219","relation":"identification"},{"oeis_id":"A292219","relation":"match_triangle"}],"bfiles":[{"node_id":"n-66d7d2fd3a2f1f0d","oeis_id":"A292219","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":37575}],"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:            6,            1;\n   3:           60,           20,            1;\n   4:          840,          420,           42,           1;\n   5:        15120,        10080,         1512,          72,          1;\n   6:       332640,       277200,        55440,        3960,        110,         1;\n   7:      8648640,      8648640,      2162160,      205920,       8580,       156,       1;\n   8:    259459200,    302702400,     90810720,    10810800,     600600,     16380,     210,     1;\n   9:   8821612800,  11762150400,   4116752640,   588107520,   40840800,   1485120,   28560,   272,   1;\n  10: 335221286400, 502831929600, 201132771840, 33522128640, 2793510720, 126977760, 3255840, 46512, 342, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+x)*P(n-1,x) + (24+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[(6 + x)*P[n - 1, x] + (24 + 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] = (6 + x)*v[i - 1] + (24 + 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":"A292219","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:            6,            1;\n  2:           60,           20,            1;\n  3:          840,          420,           42,           1;\n  4:        15120,        10080,         1512,          72,          1;\n  5:       332640,       277200,        55440,        3960,        110,         1;\n  6:      8648640,      8648640,      2162160,      205920,       8580,       156,       1;\n  7:    259459200,    302702400,     90810720,    10810800,     600600,     16380,     210,     1;\n  8:   8821612800,  11762150400,   4116752640,   588107520,   40840800,   1485120,   28560,   272,   1;\n  9: 335221286400, 502831929600, 201132771840, 33522128640, 2793510720, 126977760, 3255840, 46512, 342, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+x)*P(n-1,x) + (24+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[(6 + x)*P[n - 1, x] + (24 + 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] = (6 + x)*v[i - 1] + (24 + 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-66d7d2fd3a2f1f0d","steps":[],"edges":[],"text":"A292219"},"headline":"A292219: Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].","chain":null}