{"id":"n-ceb547fe03cf0e38","revision":0,"first_index":1,"fingerprint":"f6-d611ddd898cc65fb","name":"one_plus_derivative of Triangle of Ward numbers T(n,k) read by rows.","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"],["4","3"],["11","40","15"],["26","235","420","105"],["57","1036","4270","5040","945"],["120","3955","30268","78750","69300","10395"],["247","13896","177765","819280","1541925","1081080","135135"],["502","46371","931920","6849535","22132110","32477445","18918900","2027025"],["1013","149644","4535916","49975640","249659410","615074460","737837100","367567200","34459425"],["2036","472219","20979816","332640308","2415813400","9000561570","17824967160","18056738700","7856748900","654729075"],["4083","1467544","93584491","2074472400","21031680670","111461350000","328321743750","541941079680","474678579375","183324141000","13749310575"],["8178","4512859","406591276","12333245925","169599249820","1228059542710","5062307009760","12268604398050","17327186726850","13359746775375","4638100767300","316234143225"],["16369","13775724","1732347786","70732503296","1291302627615","12415331608680","68809710789820","230421567776400","472867816996575","582854663090700","401195716371450","126493657290000","7905853580625"],["32752","41834595","7273636932","394615549562","9406866234948","117556134981285","851480160611080","3785929827099420","10626256961119800","18870499480157325","20616357910648500","12812753369666250","3699939475732500","213458046676875"],["65519","126584584","30202853361","2154772364656","66194031183995","1057574353279080","9801144791843405","56253608433742720","207548752808682285","499887874841355000","781251355953943875","765969692353866000","433817903529635625","115551955934415000","6190283353629375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","66","786","11348","192788","3769328","83383808","2059259808","56163653184","1676677413568","54382582971328","1904328098775040","71604040163981824","2877398525487444480"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","3"]],"recurrence":{"denominator":{"coeffs":[["-1","-2"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","4","2","-2","0"],["-2","-1","5","0","-2"],["0","-2","0","3","0"],["0","0","-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","2","0","-4","-2"],["0","-1","0","4","3","0"],["0","0","-1","-1","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-9","-4","2"],["3","1","-5","0"],["0","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-3","-2"],["0","1","1","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186494,"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-82a01f79c2604456"]},{"id":186495,"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-82a01f79c2604456"]},{"id":186496,"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-82a01f79c2604456"]}],"region_exact":{},"incoming_edges":[{"id":"e-82a01f79c2604456","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-ceb547fe03cf0e38","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"b3a3c8d1489001e3afd8426675ffa1b7a47b26aea097c1ebd0c616e5698526b4","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-3fea9ad1525b48eb","role":"sequence","name":"Triangle of Ward numbers T(n,k) read by rows.","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:    4,      3;\n   3:   11,     40,       15;\n   4:   26,    235,      420,       105;\n   5:   57,   1036,     4270,      5040,        945;\n   6:  120,   3955,    30268,     78750,      69300,      10395;\n   7:  247,  13896,   177765,    819280,    1541925,    1081080,      135135;\n   8:  502,  46371,   931920,   6849535,   22132110,   32477445,    18918900,     2027025;\n   9: 1013, 149644,  4535916,  49975640,  249659410,  615074460,   737837100,   367567200,   34459425;\n  10: 2036, 472219, 20979816, 332640308, 2415813400, 9000561570, 17824967160, 18056738700, 7856748900, 654729075;\n","recurrence_sentence":"The row polynomials satisfy (n-1-2*x)*P(n,x) = -(2*n+(2*n^2+n-4)*x+(n^3-5*n-2)*x^2-(3*n^2-2)*x^3+2*n*x^4)*P(n-2,x) - (n*x+(n^2-2)*x^2+(n^2-4*n)*x^3-(3*n-4)*x^4+2*x^5)*d/dx P(n-2,x) + (3*n-2+(2*n^2+n-9)*x-(5*n+4)*x^2+2*x^3)*P(n-1,x) + ((n-1)*x+(n-3)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (2*n^2 + n - 4)*x + (n^3 - 5*n - 2)*x^2 - (3*n^2 - 2)*x^3 + 2*n*x^4)*P[n - 2, x] - (n*x + (n^2 - 2)*x^2 + (n^2 - 4*n)*x^3 - (3*n - 4)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (3*n - 2 + (2*n^2 + n - 9)*x - (5*n + 4)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 1)*x + (n - 3)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 1 - 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 3*x; for(n = 3, N, my(i = n); v[i] = (-(2*n + (2*n^2 + n - 4)*x + (n^3 - 5*n - 2)*x^2 - (3*n^2 - 2)*x^3 + 2*n*x^4)*v[i - 2] - (n*x + (n^2 - 2)*x^2 + (n^2 - 4*n)*x^3 - (3*n - 4)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (3*n - 2 + (2*n^2 + n - 9)*x - (5*n + 4)*x^2 + 2*x^3)*v[i - 1] + ((n - 1)*x + (n - 3)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 1 - 2*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-3fea9ad1525b48eb","steps":["D+1"],"edges":["e-82a01f79c2604456"],"text":"A134991 \u2192 D+1"},"headline":"A134991 \u2192 D+1","chain":null}