{"id":"n-875f8968dcda360b","revision":0,"first_index":1,"fingerprint":"f6-814590f13101ba28","name":"derivative of Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.","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","2"],["21","24","3"],["85","210","66","4"],["341","1640","975","140","5"],["1365","12162","12210","3120","255","6"],["5461","87864","140343","56840","7980","420","7"],["21845","625970","1535226","926044","198730","17556","644","8"],["87381","4425480","16291215","14085540","4337235","572040","34650","936","9"],["349525","31153122","169550370","204833640","87002685","16073442","1429680","63000","1305","10"],["1398101","218770904","1742233383","2888904480","1648085010","409798620","50205309","3208920","107415","1760","11"],["5592405","1534192530","17750490186","39878736084","29980490760","9763875792","1582615188","137600496","6617655","173910","2310","12"],["22369621","10750532520","179806190655","542090889340","529536272265","221490228960","46208722560","5248715472","340094898","12744160","269841","2964","13"],["89478485","75298466882","1814187705330","7286923148960","9150193967915","4843757876958","1274997163440","184027855440","15427462050","772952180","23193170","404040","3731","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","7","48","365","3101","29118","298915","3326023","39834486","510456779","6962713913","100640387328","1535507033269","24645699678595"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-4","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-3"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179607,"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-8130a131d1dbdd23"]},{"id":179608,"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-8130a131d1dbdd23"]},{"id":179609,"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-8130a131d1dbdd23"]},{"id":179610,"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-8130a131d1dbdd23"]}],"region_exact":{},"incoming_edges":[{"id":"e-8130a131d1dbdd23","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-875f8968dcda360b","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":"8c3dfd6ec8f2133599737519c3d1b562bbbb6ae4542910e0962975c44f79658a","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-cab14672969f6507","role":"sequence","name":"Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, 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:     0;\n   2:     1;\n   3:     5,       2;\n   4:    21,      24,        3;\n   5:    85,     210,       66,        4;\n   6:   341,    1640,      975,      140,       5;\n   7:  1365,   12162,    12210,     3120,     255,      6;\n   8:  5461,   87864,   140343,    56840,    7980,    420,     7;\n   9: 21845,  625970,  1535226,   926044,  198730,  17556,   644,   8;\n  10: 87381, 4425480, 16291215, 14085540, 4337235, 572040, 34650, 936, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4+2*x+x^2)*P(n-2,x) - (3*x+3*x^2)*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + 3*x*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[-(4 + 2*x + x^2)*P[n - 2, x] - (3*x + 3*x^2)*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + 3*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] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(4 + 2*x + x^2)*v[i - 2] - (3*x + 3*x^2)*deriv(v[i - 2], x) + (5 + 2*x)*v[i - 1] + 3*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-cab14672969f6507","steps":["D"],"edges":["e-8130a131d1dbdd23"],"text":"A111577 \u2192 D"},"headline":"A111577 \u2192 D","chain":null}