{"id":"n-95116ccff4ef466d","revision":0,"first_index":1,"fingerprint":"f6-d13aacdb3a9cb540","name":"theta_plus_one_squared of basis_map_A013609 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","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","8"],["11","64","36"],["45","408","540","128"],["197","2432","5256","3072","400"],["903","14120","42840","42240","14000","1152"],["4279","81024","318060","450560","258000","55296","3136"],["20793","462392","2232468","4161920","3486000","1314432","197568","8192"],["103049","2632192","15102864","35080192","39205600","21934080","5883136","655360","20736"],["518859","14969160","99578160","277670400","390852000","295094016","118804224","23887872","2052864","51200"],["2646723","85114560","644325300","2100019200","3579660000","3435264000","1884140160","574095360","89890560","6144000","123904"],["13648869","484094424","4110188940","15346320000","30793620000","36055514880","25332827520","10599874560","2535287040","318208000","17718272","294912"],["71039373","2754744192","25929675288","109188879360","252478710000","350172979200","302145634560","163045048320","53918472960","10407936000","1071273984","49545216","692224"],["372693519","15686015592","162137915784","760478813952","1993134171600","3202342805376","3291071976192","2193429528576","940929530112","252602521600","40220477440","3458138112","134983680","1605632"],["1968801519","89382048000","1006562285820","5205280604160","15260315070000","27915540516864","33390898597056","26635873026048","14183098465536","4964396236800","1104964596736","147710803968","10771697664","359661568","3686400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","111","1121","11357","115255","1170355","11883765","120617209","1223478755","12401423767","125607597417","1271234630677","12856001177167","129917126098139"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["4","0","-16","0","0"],["4","-20","56","16","0"],["0","-20","48","-64","16"],["0","0","-24","-32","-16"],["0","0","0","48","-16"],["0","0","0","0","16"]]},"terms":[{"coeff":{"coeffs":[["-4","-40","16","-208","0","-64"],["20","32","232","-8","320","96"],["0","-72","48","256","-112","96"],["0","0","-168","-312","-592","-192"],["0","0","0","208","304","-32"],["0","0","0","0","80","96"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","0","6","0","-32","0","-32"],["0","-12","10","24","-48","96","32"],["0","0","-12","-8","32","32","64"],["0","0","0","48","16","-192","-64"],["0","0","0","0","16","0","-64"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183198,"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-26bbf0c87904b412"]},{"id":183199,"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-26bbf0c87904b412"]},{"id":183200,"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-26bbf0c87904b412"]},{"id":183201,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-26bbf0c87904b412"]}],"region_exact":{},"incoming_edges":[{"id":"e-26bbf0c87904b412","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-95116ccff4ef466d","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"7da49d8009d8f7d5688a2c53389b931388db4441a61664e7515fdab9cb10363b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4d7e6ff489a3ffc4","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","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,        8;\n   3:     11,       64,       36;\n   4:     45,      408,      540,       128;\n   5:    197,     2432,     5256,      3072,       400;\n   6:    903,    14120,    42840,     42240,     14000,      1152;\n   7:   4279,    81024,   318060,    450560,    258000,     55296,      3136;\n   8:  20793,   462392,  2232468,   4161920,   3486000,   1314432,    197568,     8192;\n   9: 103049,  2632192, 15102864,  35080192,  39205600,  21934080,   5883136,   655360,   20736;\n  10: 518859, 14969160, 99578160, 277670400, 390852000, 295094016, 118804224, 23887872, 2052864, 51200;\n","recurrence_sentence":"The row polynomials satisfy (4*n+4-(20*n^2+20*n)*x-(24*n^3-48*n^2-56*n+16)*x^2+(48*n^4-32*n^3-64*n^2+16*n)*x^3+(16*n^5-16*n^4-16*n^3+16*n^2)*x^4)*P(n,x) = (20*n-4-(72*n^2-32*n+40)*x-(168*n^3-48*n^2-232*n-16)*x^2+(208*n^4-312*n^3+256*n^2-8*n-208)*x^3+(80*n^5+304*n^4-592*n^3-112*n^2+320*n)*x^4+(96*n^5-32*n^4-192*n^3+96*n^2+96*n-64)*x^5)*P(n-1,x) + (1-12*n*x-(12*n^2-10*n-6)*x^2+(48*n^3-8*n^2+24*n)*x^3+(16*n^4+16*n^3+32*n^2-48*n-32)*x^4-(192*n^3-32*n^2-96*n)*x^5-(64*n^4+64*n^3-64*n^2-32*n+32)*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[((20*n - 4 - (72*n^2 - 32*n + 40)*x - (168*n^3 - 48*n^2 - 232*n - 16)*x^2 + (208*n^4 - 312*n^3 + 256*n^2 - 8*n - 208)*x^3 + (80*n^5 + 304*n^4 - 592*n^3 - 112*n^2 + 320*n)*x^4 + (96*n^5 - 32*n^4 - 192*n^3 + 96*n^2 + 96*n - 64)*x^5)*P[n - 1, x] + (1 - 12*n*x - (12*n^2 - 10*n - 6)*x^2 + (48*n^3 - 8*n^2 + 24*n)*x^3 + (16*n^4 + 16*n^3 + 32*n^2 - 48*n - 32)*x^4 - (192*n^3 - 32*n^2 - 96*n)*x^5 - (64*n^4 + 64*n^3 - 64*n^2 - 32*n + 32)*x^6)*D[P[n - 1, x], x])/(4*n + 4 - (20*n^2 + 20*n)*x - (24*n^3 - 48*n^2 - 56*n + 16)*x^2 + (48*n^4 - 32*n^3 - 64*n^2 + 16*n)*x^3 + (16*n^5 - 16*n^4 - 16*n^3 + 16*n^2)*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] = ((20*n - 4 - (72*n^2 - 32*n + 40)*x - (168*n^3 - 48*n^2 - 232*n - 16)*x^2 + (208*n^4 - 312*n^3 + 256*n^2 - 8*n - 208)*x^3 + (80*n^5 + 304*n^4 - 592*n^3 - 112*n^2 + 320*n)*x^4 + (96*n^5 - 32*n^4 - 192*n^3 + 96*n^2 + 96*n - 64)*x^5)*v[i - 1] + (1 - 12*n*x - (12*n^2 - 10*n - 6)*x^2 + (48*n^3 - 8*n^2 + 24*n)*x^3 + (16*n^4 + 16*n^3 + 32*n^2 - 48*n - 32)*x^4 - (192*n^3 - 32*n^2 - 96*n)*x^5 - (64*n^4 + 64*n^3 - 64*n^2 - 32*n + 32)*x^6)*deriv(v[i - 1], x))/(4*n + 4 - (20*n^2 + 20*n)*x - (24*n^3 - 48*n^2 - 56*n + 16)*x^2 + (48*n^4 - 32*n^3 - 64*n^2 + 16*n)*x^3 + (16*n^5 - 16*n^4 - 16*n^3 + 16*n^2)*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-0d75f6272d2a2a84","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-077b645e97fdde1b","e-26bbf0c87904b412"],"text":"A114656 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A114656 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}