{"id":"n-85d5c744baea07e7","revision":1,"first_index":0,"fingerprint":"f6-5982380e7440ea27","name":"derivative of Triangle read by rows: T(n,k) = binomial(k,n-k).","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"],["2","2"],["1","6","3"],["0","6","12","4"],["0","2","18","20","5"],["0","0","12","40","30","6"],["0","0","3","40","75","42","7"],["0","0","0","20","100","126","56","8"],["0","0","0","4","75","210","196","72","9"],["0","0","0","0","30","210","392","288","90","10"],["0","0","0","0","5","126","490","672","405","110","11"],["0","0","0","0","0","42","392","1008","1080","550","132","12"],["0","0","0","0","0","6","196","1008","1890","1650","726","156","13"],["0","0","0","0","0","0","56","672","2268","3300","2420","936","182","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","4","10","22","45","88","167","310","566","1020","1819","3216","5645","9848"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["2","2"],["-1","0"],["-2","-2"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["16","4","0"],["-28","-24","-6"],["14","8","0"],["-2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","2","0"],["0","-12","-11","-2"],["0","4","1","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160860,"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-b66135a965ab5be8"]},{"id":160861,"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-b66135a965ab5be8"]},{"id":160862,"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-b66135a965ab5be8"]}],"region_exact":{},"incoming_edges":[{"id":"e-b66135a965ab5be8","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-85d5c744baea07e7","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":"7a74702930bdae0af5d44ef389cdca45a2a7ad3f0ce66498cbf4d6932140ad47","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-21854f4257effb01","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(k,n-k).","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":[{"id":"l-17b1973f390863be","kind":"oeis_match","node_a_id":"n-85d5c744baea07e7","node_b_id":null,"fingerprint":null,"oeis_id":"A004798","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A004798","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A004798","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-85d5c744baea07e7","oeis_id":"A004798","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":36,"oeis_terms_available":36,"term_count":200,"raw_size":5552}],"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: 2, 2;\n  3: 1, 6,  3;\n  4: 0, 6, 12,  4;\n  5: 0, 2, 18, 20,   5;\n  6: 0, 0, 12, 40,  30,   6;\n  7: 0, 0,  3, 40,  75,  42,   7;\n  8: 0, 0,  0, 20, 100, 126,  56,  8;\n  9: 0, 0,  0,  4,  75, 210, 196, 72, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^3-2*n^2-n+2-(2*n^2-2)*x)*P(n,x) = -(2*n^3-14*n^2+28*n-16-(8*n^2-24*n+4)*x+6*n*x^2)*P(n-1,x) + ((4*n^2-12*n+8)*x+(n^2-11*n+2)*x^2-2*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 14*n^2 + 28*n - 16 - (8*n^2 - 24*n + 4)*x + 6*n*x^2)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (n^2 - 11*n + 2)*x^2 - 2*n*x^3)*D[P[n - 1, x], x])/(n^3 - 2*n^2 - n + 2 - (2*n^2 - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 14*n^2 + 28*n - 16 - (8*n^2 - 24*n + 4)*x + 6*n*x^2)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (n^2 - 11*n + 2)*x^2 - 2*n*x^3)*deriv(v[i - 1], x))/(n^3 - 2*n^2 - n + 2 - (2*n^2 - 2)*x)); 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-21854f4257effb01","steps":["D"],"edges":["e-b66135a965ab5be8"],"text":"A030528 \u2192 D"},"headline":"A030528 \u2192 D","chain":null}
