{"id":"n-32fce1bd1e1cbfb9","revision":0,"first_index":0,"fingerprint":"f6-ec24c0ec2e60f511","name":"derivative of Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, 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"],["8","2"],["31","62","3"],["102","652","306","4"],["317","4812","7218","1268","5"],["964","30174","103008","60348","4820","6"],["2907","173346","1142841","1523788","433365","17442","7"],["8738","946168","10950618","27702872","18251030","2838504","61166","8"],["26233","5004608","95624640","412982488","516228110","191249280","17516128","209864","9"],["78720","25949914","784814664","5403537800","11426275600","8105306700","1831234216","103799656","708480","10"],["236183","132845862","6170439279","64583132520","214528977070","257434772484","113020481910","16454504744","597806379","2361830","11"],["708574","674463876","47057031762","723204322108","3584261327020","6720727146024","5017965857828","1446408644216","141171095286","3372319380","7794314","12"],["2125749","3405858516","351043372014","7717404875020","54956424273315","152401026329640","177801197384580","87930278837304","17364160968795","1170144573380","18732221838","25508988","13"],["6377276","17138414326","2576829114528","79411343709604","789617043593460","3110926624645794","5349717618211456","4147902166194392","1421310678468228","198528359274010","9448373419936","102830485956","82904588","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","10","96","1064","13620","199320","3293696","60759104","1238841360","27681705760","672925558272","17684850710400","499713846329152","15109559094813568"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["8","2"],["31","62","3"]],"recurrence":{"denominator":{"coeffs":[["-1"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0","-4"],["0","0","-8"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-4","12"],["0","-8","32"],["0","0","-16"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","4"],["0","0","-8","8"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-3","6","5"],["-6","24","-22"],["0","-8","28"],["0","0","-8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-2","0","2"],["0","-4","12","-8"],["0","0","-8","8"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","6"],["8","-16"],["0","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","2"],["0","4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183907,"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-33ec896b8507b31a"]},{"id":183908,"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-33ec896b8507b31a"]},{"id":183909,"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-33ec896b8507b31a"]},{"id":183910,"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-33ec896b8507b31a"]}],"region_exact":{},"incoming_edges":[{"id":"e-33ec896b8507b31a","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-32fce1bd1e1cbfb9","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":"7e38b14b4e4c66e585af95da2a410922d8815f2de297a4343f3ba52a9aaec22b","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-14be0ce6edbde2a1","role":"sequence","name":"Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows.","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:     0;\n  1:     1;\n  2:     8,       2;\n  3:    31,      62,        3;\n  4:   102,     652,      306,         4;\n  5:   317,    4812,     7218,      1268,         5;\n  6:   964,   30174,   103008,     60348,      4820,         6;\n  7:  2907,  173346,  1142841,   1523788,    433365,     17442,        7;\n  8:  8738,  946168, 10950618,  27702872,  18251030,   2838504,    61166,      8;\n  9: 26233, 5004608, 95624640, 412982488, 516228110, 191249280, 17516128, 209864, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = -(8*n+4)*x^2*P(n-4,x) - ((8*n+4)*x+(16*n^2-32*n-12)*x^2)*P(n-3,x) - ((8*n+4)*x^2-(8*n+4)*x^3)*d/dx P(n-3,x) - (6*n+3+(8*n^2-24*n-6)*x+(8*n^3-28*n^2+22*n-5)*x^2)*P(n-2,x) - ((4*n+2)*x+(8*n^2-12*n)*x^2-(8*n^2-8*n+2)*x^3)*d/dx P(n-2,x) + (8*n-2+(8*n^2-16*n+6)*x)*P(n-1,x) + ((4*n-2)*x-(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 2*x, P(3,x) = 31 + 62*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 8 + 2*x;\nP[3, x_] := 31 + 62*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 4)*x^2*P[n - 4, x] - ((8*n + 4)*x + (16*n^2 - 32*n - 12)*x^2)*P[n - 3, x] - ((8*n + 4)*x^2 - (8*n + 4)*x^3)*D[P[n - 3, x], x] - (6*n + 3 + (8*n^2 - 24*n - 6)*x + (8*n^3 - 28*n^2 + 22*n - 5)*x^2)*P[n - 2, x] - ((4*n + 2)*x + (8*n^2 - 12*n)*x^2 - (8*n^2 - 8*n + 2)*x^3)*D[P[n - 2, x], x] + (8*n - 2 + (8*n^2 - 16*n + 6)*x)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 8 + 2*x; v[4] = 31 + 62*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n + 4)*x^2*v[i - 4] - ((8*n + 4)*x + (16*n^2 - 32*n - 12)*x^2)*v[i - 3] - ((8*n + 4)*x^2 - (8*n + 4)*x^3)*deriv(v[i - 3], x) - (6*n + 3 + (8*n^2 - 24*n - 6)*x + (8*n^3 - 28*n^2 + 22*n - 5)*x^2)*v[i - 2] - ((4*n + 2)*x + (8*n^2 - 12*n)*x^2 - (8*n^2 - 8*n + 2)*x^3)*deriv(v[i - 2], x) + (8*n - 2 + (8*n^2 - 16*n + 6)*x)*v[i - 1] + ((4*n - 2)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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-14be0ce6edbde2a1","steps":["D"],"edges":["e-33ec896b8507b31a"],"text":"A144439 \u2192 D"},"headline":"A144439 \u2192 D","chain":null}