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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b35c340016290f98"]},{"id":193177,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-b35c340016290f98"]},{"id":193178,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve 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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"617168f80a188795389aee2dd2fda798d6f0725e06716738721db188155e755e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2ff1c5d05cc7026f","role":"sequence","name":"basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        1;\n  1:        6,         84,         162;\n  2:       36,       1008,        5913,        12096,          8100;\n  3:      216,       9072,       88938,       365904,        741150,         734832,          285768;\n  4:     1296,      72576,      997272,      6168960,      20900025,       41640480,        48866328,         31352832,          8503056;\n  5:     7776,     544320,     9622800,     79470720,     377682750,     1131702516,      2220774570,       2860945920,       2338340400,       1102248000,         228637728;\n  6:    46656,    3919104,    84715632,    875266560,    5354869500,    21429905616,     58997125089,     114292829952,     156147994620,     147701232000,       92255323248,       34284321792,        5748065856;\n  7:   279936,   27433728,   701029728,   8696491776,   65084682600,   325958983056,   1155440682018,    2989720819008,    5730042565926,    8148974576400,     8505266322168,     6339742504704,     3198798648864,      979960197888,      137749507200;\n  8:  1679616,  188116992,  5547771648,  80325955584,  710913218400,  4276122617088,  18545825959920,   60061099650048,  148221684433809,  281536404609600,   412171519803120,   461821242645504,   389267641867104,   239110288284672,   101108138284800,   26330359136256,    3184768606464;\n  9: 10077696, 1269789696, 42447255552, 702616965120, 7178563310400, 50407952628096, 258581812243104, 1005888033690624, 3037474464607926, 7221221266562100, 13612385563613298, 20369232682235136, 24079771617658848, 22229907108990336, 15699517959844800, 8195324281159680, 2980943415650304, 674818305831936, 71607703822848;\n","recurrence_sentence":"The row polynomials satisfy (4+(42*n-14)*x+(49*n^2+96*n-72)*x^2+(168*n^2-42*n-42)*x^3+(144*n^2-144*n+36)*x^4)*P(n,x) = (24+(252*n+252)*x+(294*n^2+2046*n+1098)*x^2+(2037*n^2+6594*n+2541)*x^3+(5274*n^2+10530*n+3294)*x^4+(6048*n^2+8316*n+2268)*x^5+(2592*n^2+2592*n+648)*x^6)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((24 + (252*n + 252)*x + (294*n^2 + 2046*n + 1098)*x^2 + (2037*n^2 + 6594*n + 2541)*x^3 + (5274*n^2 + 10530*n + 3294)*x^4 + (6048*n^2 + 8316*n + 2268)*x^5 + (2592*n^2 + 2592*n + 648)*x^6)*P[n - 1, x])/(4 + (42*n - 14)*x + (49*n^2 + 96*n - 72)*x^2 + (168*n^2 - 42*n - 42)*x^3 + (144*n^2 - 144*n + 36)*x^4)]];\nTable[CoefficientList[P[n, x], x], {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] = ((24 + (252*n + 252)*x + (294*n^2 + 2046*n + 1098)*x^2 + (2037*n^2 + 6594*n + 2541)*x^3 + (5274*n^2 + 10530*n + 3294)*x^4 + (6048*n^2 + 8316*n + 2268)*x^5 + (2592*n^2 + 2592*n + 648)*x^6)*v[i - 1])/(4 + (42*n - 14)*x + (49*n^2 + 96*n - 72)*x^2 + (168*n^2 - 42*n - 42)*x^3 + (144*n^2 - 144*n + 36)*x^4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-eed4f744ce063213","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-bf76d3591eaf645b","e-b35c340016290f98"],"text":"A200536 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A200536 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}