{"id":"n-f09a1f04142f8fd5","revision":0,"first_index":1,"fingerprint":"f6-304b7b37ebb5392c","name":"one_plus_derivative of basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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"],["3","2"],["8","14","4"],["17","54","44","8"],["35","158","232","120","16"],["68","404","868","808","304","32"],["129","948","2696","3704","2496","736","64"],["239","2098","7444","13624","13584","7136","1728","128"],["436","4450","18944","43464","57936","45024","19328","3968","256"],["785","9138","45412","125480","210336","218880","138816","50304","8960","512"],["1399","18294","103988","336408","680944","892800","758528","405376","126976","19968","1024"],["2472","35880","229656","851824","2023216","3209120","3438784","2461312","1134848","312832","44032","2048"],["4337","69192","492528","2061104","5623344","10483360","13606272","12288384","7583488","3071488","755712","96256","4096"],["7563","131554","1030968","4805872","14816736","31775360","48529216","53124224","41369600","22408192","8086528","1796096","208896","8192"],["13120","247126","2114444","10866480","37369392","90675840","159483008","205603712","194141440","132668416","63969280","20805632","4210688","450560","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","123","561","2484","10773","45981","193806","808623","3345705","13746024","56139561","228098997","922635522"],"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":[["14","68","8","-16"],["-2","-100","-24","16"],["-20","32","16","0"],["8","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["30","-248","-64","96","32"],["-184","144","128","0","0"],["112","-16","-32","0","0"],["-16","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","-38","-56","144","144","32"],["-30","8","208","160","32","0"],["20","48","16","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175331,"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-6a13a5990ade516d"]},{"id":175332,"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-6a13a5990ade516d"]},{"id":175333,"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-6a13a5990ade516d"]}],"region_exact":{},"incoming_edges":[{"id":"e-6a13a5990ade516d","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-f09a1f04142f8fd5","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":"3c649090335d27d4f99aa708b3c99ceb434dd26d5c1c942066df8094d119a39e","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-273f45fc84d8c52d","role":"sequence","name":"basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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:   1;\n   2:   3,    2;\n   3:   8,   14,     4;\n   4:  17,   54,    44,      8;\n   5:  35,  158,   232,    120,     16;\n   6:  68,  404,   868,    808,    304,     32;\n   7: 129,  948,  2696,   3704,   2496,    736,     64;\n   8: 239, 2098,  7444,  13624,  13584,   7136,   1728,   128;\n   9: 436, 4450, 18944,  43464,  57936,  45024,  19328,  3968,  256;\n  10: 785, 9138, 45412, 125480, 210336, 218880, 138816, 50304, 8960, 512;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-20*n^2-2*n+14+(32*n^2-100*n+68)*x+(16*n^2-24*n+8)*x^2+(16*n-16)*x^3)*P(n,x) = -(16*n^3-112*n^2+184*n-30+(16*n^2-144*n+248)*x+(32*n^2-128*n+64)*x^2-96*x^3-32*x^4)*P(n-1,x) + (20*n^2-30*n+5+(48*n^2+8*n-38)*x+(16*n^2+208*n-56)*x^2+(160*n+144)*x^3+(32*n+144)*x^4+32*x^5)*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[(-(16*n^3 - 112*n^2 + 184*n - 30 + (16*n^2 - 144*n + 248)*x + (32*n^2 - 128*n + 64)*x^2 - 96*x^3 - 32*x^4)*P[n - 1, x] + (20*n^2 - 30*n + 5 + (48*n^2 + 8*n - 38)*x + (16*n^2 + 208*n - 56)*x^2 + (160*n + 144)*x^3 + (32*n + 144)*x^4 + 32*x^5)*D[P[n - 1, x], x])/(8*n^3 - 20*n^2 - 2*n + 14 + (32*n^2 - 100*n + 68)*x + (16*n^2 - 24*n + 8)*x^2 + (16*n - 16)*x^3)]];\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] = (-(16*n^3 - 112*n^2 + 184*n - 30 + (16*n^2 - 144*n + 248)*x + (32*n^2 - 128*n + 64)*x^2 - 96*x^3 - 32*x^4)*v[i - 1] + (20*n^2 - 30*n + 5 + (48*n^2 + 8*n - 38)*x + (16*n^2 + 208*n - 56)*x^2 + (160*n + 144)*x^3 + (32*n + 144)*x^4 + 32*x^5)*deriv(v[i - 1], x))/(8*n^3 - 20*n^2 - 2*n + 14 + (32*n^2 - 100*n + 68)*x + (16*n^2 - 24*n + 8)*x^2 + (16*n - 16)*x^3)); 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-392f49b496f79fbf","steps":["basis A013609","D+1"],"edges":["e-6b05245c845c9a6d","e-6a13a5990ade516d"],"text":"A026729 \u2192 basis A013609 \u2192 D+1"},"headline":"A026729 \u2192 basis A013609 \u2192 D+1","chain":null}