{"id":"n-9fed93b7289c027d","revision":0,"first_index":1,"fingerprint":"f6-8df4cfbffdd1234a","name":"one_plus_derivative of shift_x_plus_one of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by 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","1"],["10","7","1"],["30","34","11","1"],["87","138","73","15","1"],["245","503","377","127","19","1"],["676","1709","1672","801","196","23","1"],["1836","5524","6698","4206","1464","280","27","1"],["4925","17204","24950","19490","8906","2420","379","31","1"],["13079","52061","88006","82490","46982","16762","3723","493","35","1"],["34446","153971","297539","325974","223208","99338","28953","5427","622","39","1"],["90090","446934","972429","1220727","978792","524280","190637","46847","7586","766","43","1"],["234227","1277310","3091287","4378057","4029003","2529996","1107765","339521","72001","10254","925","47","1"],["605865","3602867","9602871","15153921","15756937","11369355","5839941","2155563","569843","106161","13485","1099","51","1"],["1560200","10048985","29254370","50919055","59073100","48212185","28477300","12335125","3926560","911315","151262","17333","1288","55","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","18","76","314","1272","5078","20036","78306","303632","1169518","4479132","17070394","64777960","244888134"],"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":[["0","0","4","-5","-1"],["-4","-2","-11","4","1"],["4","0","7","1","0"],["0","2","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["-32","-56","20","53","14","1"],["4","66","49","-5","-2","0"],["28","4","-19","-3","0","0"],["-8","-6","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","32","120","156","83","16","1"],["16","88","138","81","16","1","0"],["16","32","18","2","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174592,"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-b80e45f3bdb18660"]},{"id":174593,"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-b80e45f3bdb18660"]},{"id":174594,"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-b80e45f3bdb18660"]}],"region_exact":{},"incoming_edges":[{"id":"e-b80e45f3bdb18660","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-9fed93b7289c027d","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":"57a8d14b9fc625564c4472f5b48c0ef39b2f8604a1d0c82636b4e1c28ef8d49c","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-85a14e0b559789f0","role":"sequence","name":"shift_x_plus_one of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by 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":[{"id":"l-cb5298d08e525bcc","kind":"fingerprint_related","node_a_id":"n-9fed93b7289c027d","node_b_id":"n-412287924c07c5fe","fingerprint":"f6-8df4cfbffdd1234a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8df4cfbffdd1234a"},"other_id":"n-412287924c07c5fe","other":{"id":"n-412287924c07c5fe","name":"one_plus_derivative of basis_map_A007318 of delannoy","first_index":0,"fingerprint":"f6-8df4cfbffdd1234a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["10","7","1"]],"card_rows":[["1"],["3","1"],["10","7","1"],["30","34","11","1"]],"row_sums_preview":["1","4","18","76","314","1272","5078","20036"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e00a1cb428776eaa","steps":["x \u21a6 x+1","D+1"],"edges":["e-41d30650643fabff","e-502d365fc3081f31"],"text":"delannoy \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"delannoy \u2192 x \u21a6 x+1 \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:     3,     1;\n   3:    10,     7,     1;\n   4:    30,    34,    11,     1;\n   5:    87,   138,    73,    15,     1;\n   6:   245,   503,   377,   127,    19,     1;\n   7:   676,  1709,  1672,   801,   196,    23,    1;\n   8:  1836,  5524,  6698,  4206,  1464,   280,   27,   1;\n   9:  4925, 17204, 24950, 19490,  8906,  2420,  379,  31,  1;\n  10: 13079, 52061, 88006, 82490, 46982, 16762, 3723, 493, 35, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-4*n+(2*n^3-2*n)*x+(7*n^2-11*n+4)*x^2+(n^2+4*n-5)*x^3+(n-1)*x^4)*P(n,x) = -(8*n^3-28*n^2-4*n+32+(6*n^3-4*n^2-66*n+56)*x+(19*n^2-49*n-20)*x^2+(3*n^2+5*n-53)*x^3+(2*n-14)*x^4-x^5)*P(n-1,x) + (16*n^2+16*n+(32*n^2+88*n+32)*x+(18*n^2+138*n+120)*x^2+(2*n^2+81*n+156)*x^3+(16*n+83)*x^4+(n+16)*x^5+x^6)*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 - 4*n + 32 + (6*n^3 - 4*n^2 - 66*n + 56)*x + (19*n^2 - 49*n - 20)*x^2 + (3*n^2 + 5*n - 53)*x^3 + (2*n - 14)*x^4 - x^5)*P[n - 1, x] + (16*n^2 + 16*n + (32*n^2 + 88*n + 32)*x + (18*n^2 + 138*n + 120)*x^2 + (2*n^2 + 81*n + 156)*x^3 + (16*n + 83)*x^4 + (n + 16)*x^5 + x^6)*D[P[n - 1, x], x])/(4*n^2 - 4*n + (2*n^3 - 2*n)*x + (7*n^2 - 11*n + 4)*x^2 + (n^2 + 4*n - 5)*x^3 + (n - 1)*x^4)]];\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 - 4*n + 32 + (6*n^3 - 4*n^2 - 66*n + 56)*x + (19*n^2 - 49*n - 20)*x^2 + (3*n^2 + 5*n - 53)*x^3 + (2*n - 14)*x^4 - x^5)*v[i - 1] + (16*n^2 + 16*n + (32*n^2 + 88*n + 32)*x + (18*n^2 + 138*n + 120)*x^2 + (2*n^2 + 81*n + 156)*x^3 + (16*n + 83)*x^4 + (n + 16)*x^5 + x^6)*deriv(v[i - 1], x))/(4*n^2 - 4*n + (2*n^3 - 2*n)*x + (7*n^2 - 11*n + 4)*x^2 + (n^2 + 4*n - 5)*x^3 + (n - 1)*x^4)); 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-811c78acf96cbc19","steps":["basis A007318","D+1"],"edges":["e-0985ee5bc77b64e0","e-b80e45f3bdb18660"],"text":"A008288 \u2192 basis A007318 \u2192 D+1"},"headline":"A008288 \u2192 basis A007318 \u2192 D+1","chain":null}
