{"id":"n-1591149e0e86f5c2","revision":1,"first_index":1,"fingerprint":"f6-0c5d56a5621af424","name":"theta_plus_one of Triangle of B-analogs of Stirling numbers of the second kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","8","3"],["1","26","27","4"],["1","80","174","64","5"],["1","242","990","680","125","6"],["1","728","5313","6080","1975","216","7"],["1","2186","27657","49644","25375","4746","343","8"],["1","6560","141564","384384","290430","82656","9996","512","9"],["1","19682","717660","2879440","3094350","1257732","226380","19104","729","10"],["1","59048","3617823","21112960","31448450","17548272","4410294","545280","33885","1000","11"],["1","177146","18177687","152614484","309427250","230769132","77806806","13219536","1189485","56650","1331","12"],["1","531440","91154154","1092538304","2975613355","2909773152","1280719132","287215104","35093223","2398000","90266","1728","13"],["1","1594322","456567930","7769307000","28146193075","35578240698","20044084060","5771905568","919701783","84554470","4533386","138216","2197","14"],["1","4782968","2285231133","54993906240","263027371425","425136079368","302081040261","109486108160","22128324075","2628425800","188211023","8124480","204659","2744","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","58","324","2044","14320","109960","916112","8215088","78777024","803439520","8675127872","98776822720","1181967812352"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165573,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 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-694ead71eec92686"]},{"id":165574,"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-694ead71eec92686"]},{"id":165575,"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-694ead71eec92686"]},{"id":165576,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-694ead71eec92686"]}],"region_exact":{},"incoming_edges":[{"id":"e-694ead71eec92686","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-1591149e0e86f5c2","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":"7202355af38c853e54f45762847e00e36ad60344d20c45c38cdcc3c63d3351f6","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":"unordered"}],"premises":[{"position":0,"node_id":"n-d55b2cfe05a04774","role":"sequence","name":"Triangle of B-analogs of Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":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: 1,     2;\n   3: 1,     8,      3;\n   4: 1,    26,     27,       4;\n   5: 1,    80,    174,      64,       5;\n   6: 1,   242,    990,     680,     125,       6;\n   7: 1,   728,   5313,    6080,    1975,     216,      7;\n   8: 1,  2186,  27657,   49644,   25375,    4746,    343,     8;\n   9: 1,  6560, 141564,  384384,  290430,   82656,   9996,   512,   9;\n  10: 1, 19682, 717660, 2879440, 3094350, 1257732, 226380, 19104, 729, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x-x^2)*P(n-2,x) + (2*x-2*x^2)*d/dx P(n-2,x) + 2*x*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x - x^2)*P[n - 2, x] + (2*x - 2*x^2)*D[P[n - 2, x], x] + 2*x*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (1 + 2*x - x^2)*v[i - 2] + (2*x - 2*x^2)*deriv(v[i - 2], x) + 2*x*v[i - 1] + 2*x*deriv(v[i - 1], 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}}],"derivation":{"root":"n-d55b2cfe05a04774","steps":["\u03b8+1"],"edges":["e-694ead71eec92686"],"text":"A039755 \u2192 \u03b8+1"},"headline":"A039755 \u2192 \u03b8+1","chain":null}