{"id":"n-a33841ee0ddb033c","revision":0,"first_index":1,"fingerprint":"f6-d672f84bcb723328","name":"one_plus_derivative of reversal 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":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","3"],["6","15","5"],["8","35","35","7"],["10","63","117","66","9"],["12","99","275","297","110","11"],["14","143","533","884","637","169","13"],["16","195","915","2075","2340","1218","245","15"],["18","255","1445","4182","6545","5406","2142","340","17"],["20","323","2147","7581","15314","17765","11286","3534","456","19"],["22","399","3045","12712","31605","47775","43043","21780","5544","595","21"],["24","483","4163","20079","59432","111412","131859","95381","39468","8349","759","23"],["26","575","5525","30250","104025","233580","345100","330200","196625","67925","12155","950","25"],["28","675","7155","43857","171990","450927","802332","963900","763776","381810","111969","17199","1170","27"],["30","783","9077","61596","271469","815045","1700937","2472888","2472888","1653522","704990","177944","23751","1421","29"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","26","85","265","804","2393","7019","20350","58445","166541","471432","1326961","3716815","10366370"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-24","51","-12"],["52","-58","8"],["-36","16","0"],["8","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","36","105","-36"],["24","36","-142","24"],["-28","-76","48","0"],["8","24","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","9","32","-16"],["0","6","16","-32","0"],["0","-4","-16","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173443,"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-b04ae3afc23765ad"]},{"id":173444,"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-b04ae3afc23765ad"]},{"id":173445,"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-b04ae3afc23765ad"]}],"region_exact":{},"incoming_edges":[{"id":"e-b04ae3afc23765ad","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-a33841ee0ddb033c","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":"1661f8e3db635a0bb8340eca0881cae2a816af0a541efab0944eeeb06fe49e27","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-7895ffdfea7c95d5","role":"sequence","name":"reversal 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":[{"id":"l-3f181d267891c1bc","kind":"fingerprint_related","node_a_id":"n-a8f047ca6c0b04d7","node_b_id":"n-a33841ee0ddb033c","fingerprint":"f6-d672f84bcb723328","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d672f84bcb723328"},"other_id":"n-a8f047ca6c0b04d7","other":{"id":"n-a8f047ca6c0b04d7","name":"one_plus_derivative of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","first_index":0,"fingerprint":"f6-d672f84bcb723328","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","3"],["6","15","5"]],"card_rows":[["1"],["4","3"],["6","15","5"],["8","35","35","7"]],"row_sums_preview":["1","7","26","85","265","804","2393","7019"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9496336e2b29e28e","steps":["D+1"],"edges":["e-96ae67f48970d85a"],"text":"A082985 \u2192 D+1"},"headline":"A082985 \u2192 D+1","chain":null}}],"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:  6,  15,    5;\n   4:  8,  35,   35,    7;\n   5: 10,  63,  117,   66,     9;\n   6: 12,  99,  275,  297,   110,    11;\n   7: 14, 143,  533,  884,   637,   169,    13;\n   8: 16, 195,  915, 2075,  2340,  1218,   245,   15;\n   9: 18, 255, 1445, 4182,  6545,  5406,  2142,  340,  17;\n  10: 20, 323, 2147, 7581, 15314, 17765, 11286, 3534, 456, 19;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-36*n^2+52*n-24+(16*n^2-58*n+51)*x+(8*n-12)*x^2)*P(n,x) = (8*n^3-28*n^2+24*n+(24*n^3-76*n^2+36*n+36)*x+(48*n^2-142*n+105)*x^2+(24*n-36)*x^3)*P(n-1,x) - ((4*n^2-6*n)*x+(16*n^2-16*n-9)*x^2+(32*n-32)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 28*n^2 + 24*n + (24*n^3 - 76*n^2 + 36*n + 36)*x + (48*n^2 - 142*n + 105)*x^2 + (24*n - 36)*x^3)*P[n - 1, x] - ((4*n^2 - 6*n)*x + (16*n^2 - 16*n - 9)*x^2 + (32*n - 32)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 36*n^2 + 52*n - 24 + (16*n^2 - 58*n + 51)*x + (8*n - 12)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 28*n^2 + 24*n + (24*n^3 - 76*n^2 + 36*n + 36)*x + (48*n^2 - 142*n + 105)*x^2 + (24*n - 36)*x^3)*v[i - 1] - ((4*n^2 - 6*n)*x + (16*n^2 - 16*n - 9)*x^2 + (32*n - 32)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 36*n^2 + 52*n - 24 + (16*n^2 - 58*n + 51)*x + (8*n - 12)*x^2)); 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":["reverse","D+1"],"edges":["e-2020a36645e8b930","e-b04ae3afc23765ad"],"text":"A111125 \u2192 reverse \u2192 D+1"},"headline":"A111125 \u2192 reverse \u2192 D+1","chain":null}
