{"id":"n-679d41c81ca02564","revision":0,"first_index":1,"fingerprint":"f6-6c6f5aa637e12da4","name":"theta_plus_one of derivative of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["5","4"],["14","28","9"],["30","108","81","16"],["55","308","396","176","25"],["91","728","1404","1040","325","36"],["140","1512","4050","4400","2250","540","49"],["204","2856","10098","14960","11050","4284","833","64"],["285","5016","22572","43472","43225","23940","7448","1216","81"],["385","8316","46332","112112","143325","105840","46648","12096","1701","100"],["506","13156","88803","263120","418600","394128","229908","83904","18630","2300","121"],["650","20020","160875","572000","1105000","1285200","949620","456000","141750","27500","3025","144"],["819","29484","277992","1166880","2685150","3767472","3418632","2068416","841995","227700","39204","3888","169"],["1015","42224","461448","2255968","6088550","10116360","11015592","8145984","4160079","1467400","350900","54288","4901","196"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","9","51","235","960","3624","12941","44349","147255","476855","1513176","4721784","14527801","44164905"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["12","12"],["-14","-32"],["4","31"],["0","-13"],["0","2"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-2","-6","-1"],["4","11","3"],["0","-7","-3"],["0","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-8","-6","-1"],["0","8","14","3"],["0","0","-12","-3"],["0","0","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174217,"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-9db6104006f19c66"]},{"id":174218,"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-9db6104006f19c66"]},{"id":174219,"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-9db6104006f19c66"]}],"region_exact":{},"incoming_edges":[{"id":"e-9db6104006f19c66","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-679d41c81ca02564","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":"c199741cd702762c93cf73214ccf85f081bf026b1fe3089c730843886024bc8e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a4555bce014ca1b5","role":"sequence","name":"derivative of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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:   0;\n   2:   1;\n   3:   5,    4;\n   4:  14,   28,     9;\n   5:  30,  108,    81,    16;\n   6:  55,  308,   396,   176,    25;\n   7:  91,  728,  1404,  1040,   325,    36;\n   8: 140, 1512,  4050,  4400,  2250,   540,   49;\n   9: 204, 2856, 10098, 14960, 11050,  4284,  833,   64;\n  10: 285, 5016, 22572, 43472, 43225, 23940, 7448, 1216, 81;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-14*n+12+(2*n^4-13*n^3+31*n^2-32*n+12)*x)*P(n,x) = (4*n^2-2*n+(2*n^4-7*n^3+11*n^2-6*n)*x+(n^4-3*n^3+3*n^2-n)*x^2)*P(n-1,x) + ((8*n-8)*x+(4*n^3-12*n^2+14*n-6)*x^2+(n^3-3*n^2+3*n-1)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n + (2*n^4 - 7*n^3 + 11*n^2 - 6*n)*x + (n^4 - 3*n^3 + 3*n^2 - n)*x^2)*P[n - 1, x] + ((8*n - 8)*x + (4*n^3 - 12*n^2 + 14*n - 6)*x^2 + (n^3 - 3*n^2 + 3*n - 1)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12 + (2*n^4 - 13*n^3 + 31*n^2 - 32*n + 12)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 2*n + (2*n^4 - 7*n^3 + 11*n^2 - 6*n)*x + (n^4 - 3*n^3 + 3*n^2 - n)*x^2)*v[i - 1] + ((8*n - 8)*x + (4*n^3 - 12*n^2 + 14*n - 6)*x^2 + (n^3 - 3*n^2 + 3*n - 1)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12 + (2*n^4 - 13*n^3 + 31*n^2 - 32*n + 12)*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-318700a428d7b97b","steps":["D","\u03b8+1"],"edges":["e-05948cabe35ede38","e-9db6104006f19c66"],"text":"A111125 \u2192 D \u2192 \u03b8+1"},"headline":"A111125 \u2192 D \u2192 \u03b8+1","chain":null}