{"id":"n-f8871b17e8bc3bb1","revision":0,"first_index":0,"fingerprint":"f6-03bb4e5ed48317f0","name":"theta_plus_one of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","8","9"],["1","20","81","72"],["1","40","381","1152","900"],["1","70","1281","8772","22500","16200"],["1","112","3486","44640","252765","583200","396900"],["1","168","8190","174784","1815165","9076104","19448100","12700800"],["1","240","17262","567904","9680445","87491232","400644468","812851200","514382400"],["1","330","33462","1603624","41625045","610235262","4993934148","21417423840","41664974400","25719120000"],["1","440","60687","4057504","151874195","3357488232","44150696793","335321856000","1366868074500","2571912000000","1556006760000"],["1","572","104247","9397960","486618275","15385924476","302677290657","3665545842672","26264515882500","102808904130000","188276817960000","112032486720000"],["1","728","171171","20239648","1402919375","60933395016","1702796417973","30647064324096","347916163576968","2379066947280000","9009280792314000","16132678087680000","9466745127840000"],["1","910","270543","41008396","3705179335","214132190766","8171891855505","207737891793288","3485409373756296","37557256820062320","247153882215042000","910508553106488000","1599879926604960000","927741022528320000"],["1","1120","413868","78884416","9087531310","680984786976","34403085224340","1188364914453888","28024447841838441","444188350424963520","4585017044932239960","29220862334102208000","105170227842467970000","181837240415550720000","104370865034436000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","18","174","2474","48824","1281104","43223312","1825635152","94448950112","5877772868352","433366829871360","37368099336158976","3726542178310927360","425657648927907235840"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["-4"],["4"]]},"terms":[{"coeff":{"coeffs":[["-4","0","0"],["-4","4","0"],["0","0","-1"],["0","-4","1"],["0","0","1"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["8","-4"],["0","0"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186847,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 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-51d651091a727e21"]},{"id":186848,"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-51d651091a727e21"]},{"id":186849,"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 real-rootedness)","region_exact":null,"edges":["e-51d651091a727e21"]},{"id":186850,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-51d651091a727e21"]}],"region_exact":{},"incoming_edges":[{"id":"e-51d651091a727e21","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-f8871b17e8bc3bb1","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"69276f2bde38d52c8137f5390c04909c537cacee7852f7fe6bd9633c1fe89af0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ecd77b030b8fab94","role":"sequence","name":"Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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: 1,   2;\n  2: 1,   8,     9;\n  3: 1,  20,    81,      72;\n  4: 1,  40,   381,    1152,      900;\n  5: 1,  70,  1281,    8772,    22500,     16200;\n  6: 1, 112,  3486,   44640,   252765,    583200,     396900;\n  7: 1, 168,  8190,  174784,  1815165,   9076104,   19448100,    12700800;\n  8: 1, 240, 17262,  567904,  9680445,  87491232,  400644468,   812851200,   514382400;\n  9: 1, 330, 33462, 1603624, 41625045, 610235262, 4993934148, 21417423840, 41664974400, 25719120000;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4)*P(n,x) = -(4*n+4+(4*n^3-4*n)*x+(n^5-n^4-n^3+n^2)*x^2)*P(n-2,x) + (8*n+(4*n^3-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n + 4 + (4*n^3 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*P[n - 2, x] + (8*n + (4*n^3 - 4*n)*x)*P[n - 1, x])/(4*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + 4 + (4*n^3 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*v[i - 2] + (8*n + (4*n^3 - 4*n)*x)*v[i - 1])/(4*n - 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-ecd77b030b8fab94","steps":["\u03b8+1"],"edges":["e-51d651091a727e21"],"text":"A128813 \u2192 \u03b8+1"},"headline":"A128813 \u2192 \u03b8+1","chain":null}