{"id":"n-5cf9c2bd32c3fc06","revision":0,"first_index":1,"fingerprint":"f6-e7fa2f49aa742843","name":"derivative of derivative 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":-3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["2"],["14","6"],["54","54","12"],["154","264","132","20"],["364","936","780","260","30"],["756","2700","3300","1800","450","42"],["1428","6732","11220","8840","3570","714","56"],["2508","15048","32604","34580","19950","6384","1064","72"],["4158","30888","84084","114660","88200","39984","10584","1512","90"],["6578","59202","197340","334880","328440","197064","73416","16560","2070","110"],["10010","107250","429000","884000","1071000","813960","399000","126000","24750","2750","132"],["14742","185328","875160","2148120","3139560","2930256","1809864","748440","204930","35640","3564","156"],["21112","307632","1691976","4870840","8430300","9441936","7127736","3697848","1320660","319000","49764","4524","182"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","2","20","120","570","2370","9048","32560","112210","374160","1215660","3867852","12095760","37283510"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["9"],["-9"],["2"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","0"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-1"],["0","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174211,"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-0e2eb94102ec0a8f"]},{"id":174212,"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-0e2eb94102ec0a8f"]},{"id":174213,"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-0e2eb94102ec0a8f"]}],"region_exact":{},"incoming_edges":[{"id":"e-0e2eb94102ec0a8f","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-5cf9c2bd32c3fc06","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":"e3b61f21f76bd5af1295ff2c458019fe18f5efac82d527a5f68d00101edc1423","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-a4555bce014ca1b5","role":"sequence","name":"derivative 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":[],"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:    0;\n   3:    2;\n   4:   14,     6;\n   5:   54,    54,    12;\n   6:  154,   264,   132,    20;\n   7:  364,   936,   780,   260,    30;\n   8:  756,  2700,  3300,  1800,   450,   42;\n   9: 1428,  6732, 11220,  8840,  3570,  714,   56;\n  10: 2508, 15048, 32604, 34580, 19950, 6384, 1064, 72;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-9*n+9)*P(n,x) = (2*n^2+n-1+(n^2-1)*x)*P(n-1,x) + ((4*n-4)*x+(n-1)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n - 1 + (n^2 - 1)*x)*P[n - 1, x] + ((4*n - 4)*x + (n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 9)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = ((2*n^2 + n - 1 + (n^2 - 1)*x)*v[i - 1] + ((4*n - 4)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 9)); 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":["D","D"],"edges":["e-05948cabe35ede38","e-0e2eb94102ec0a8f"],"text":"A111125 \u2192 D \u2192 D"},"headline":"A111125 \u2192 D \u2192 D","chain":null}