{"id":"n-66d49a25d4809337","revision":0,"first_index":0,"fingerprint":"f6-3fbac73d16988ce1","name":"derivative of theta_plus_one of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.","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":[[],["2"],["6","6"],["8","24","12"],["12","66","72","20"],["14","102","204","140","30"],["18","180","540","600","270","42"],["20","234","900","1500","1170","420","56"],["24","348","1728","3900","4320","2436","672","72"],["26","420","2424","6780","10170","8484","3920","936","90"],["30","570","3960","13700","26190","28770","18480","6840","1350","110"],["32","660","5100","20300","46740","65436","56840","30600","9900","1760","132"],["36","846","7560","35400","97740","167538","182448","127440","56700","15510","2376","156"],["38","954","9252","48000","150840","304374","405832","362016","216000","84810","20988","2964","182"],["42","1176","12852","76020","275940","652386","1035048","1118376","827820","418110","141372","30576","3822","210"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","12","44","170","490","1650","4300","13500","33250","100000","237500","693750","1606250","4593750"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"],["6","6"],["8","24","12"],["12","66","72","20"],["14","102","204","140","30"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","9","30","45","30","9","1"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["-3","-18","-33","-18","-3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["3","9","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173334,"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-86ed5de27c1b70ca"]},{"id":173335,"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-86ed5de27c1b70ca"]},{"id":173336,"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-86ed5de27c1b70ca"]}],"region_exact":{},"incoming_edges":[{"id":"e-86ed5de27c1b70ca","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-66d49a25d4809337","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":"93aaeae158b1bc7c5f57f9b0153e219f4b66a0c160a7f9351e660fa56c72842a","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-f93ed6bfdfdae24a","role":"sequence","name":"theta_plus_one of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.","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:  2;\n  2:  6,   6;\n  3:  8,  24,   12;\n  4: 12,  66,   72,   20;\n  5: 14, 102,  204,  140,    30;\n  6: 18, 180,  540,  600,   270,   42;\n  7: 20, 234,  900, 1500,  1170,  420,   56;\n  8: 24, 348, 1728, 3900,  4320, 2436,  672,  72;\n  9: 26, 420, 2424, 6780, 10170, 8484, 3920, 936, 90;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+9*x+30*x^2+45*x^3+30*x^4+9*x^5+x^6)*P(n-6,x) - (3+18*x+33*x^2+18*x^3+3*x^4)*P(n-4,x) + (3+9*x+3*x^2)*P(n-2,x) for n >= 6, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 6 + 6*x, P(3,x) = 8 + 24*x + 12*x^2, P(4,x) = 12 + 66*x + 72*x^2 + 20*x^3, P(5,x) = 14 + 102*x + 204*x^2 + 140*x^3 + 30*x^4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 6 + 6*x;\nP[3, x_] := 8 + 24*x + 12*x^2;\nP[4, x_] := 12 + 66*x + 72*x^2 + 20*x^3;\nP[5, x_] := 14 + 102*x + 204*x^2 + 140*x^3 + 30*x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*P[n - 6, x] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*P[n - 4, x] + (3 + 9*x + 3*x^2)*P[n - 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] = 2; v[3] = 6 + 6*x; v[4] = 8 + 24*x + 12*x^2; v[5] = 12 + 66*x + 72*x^2 + 20*x^3; v[6] = 14 + 102*x + 204*x^2 + 140*x^3 + 30*x^4; for(n = 6, N, my(i = n + 1); v[i] = (1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*v[i - 6] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*v[i - 4] + (3 + 9*x + 3*x^2)*v[i - 2]); 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-4e0b19699f4cf0db","steps":["\u03b8+1","D"],"edges":["e-9d3ba3f44f61794d","e-86ed5de27c1b70ca"],"text":"A026374 \u2192 \u03b8+1 \u2192 D"},"headline":"A026374 \u2192 \u03b8+1 \u2192 D","chain":null}