{"id":"n-2ea32e7d80305e8d","revision":0,"first_index":1,"fingerprint":"f6-5b508ded3910d61e","name":"theta_plus_one_squared of basis_map_A013611 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 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":[["1"],["2","16"],["6","80","144"],["16","320","1152","1024"],["44","1152","6192","11264","6400"],["120","3904","27936","76800","89600","36864"],["328","12736","113760","418816","742400","626688","200704"],["896","40448","433152","1998848","4761600","6119424","4014080","1048576"],["2448","125952","1572480","8724480","26118400","45195264","45158400","24117248","5308416"],["6688","386304","5508864","35708928","128780800","280498176","377724928","307232768","138018816","26214400"],["18272","1170688","18772992","139223040","587507200","1543606272","2618986496","2872049664","1964113920","760217600","126877696"],["49920","3513344","62576640","522534912","2525900800","7774027776","15924658176","22014853120","20299382784","11953766400","4060086272","603979776"],["136384","10459136","204855552","1902002176","10362803200","36568498176","87816630272","146654887936","170516938752","135371161600","69909610496","21139292160","2835349504"],["372608","30925824","660616704","6750814208","40930764800","162923986944","448902995968","879334785024","1235523207168","1236952678400","860992045056","395606753280","107743281152","13153337344"],["1017984","90913792","2103409152","23460102144","156667238400","694435921920","2159565004800","4856065556480","8006153011200","9688055808000","8503850696704","5264891707392","2177548419072","539286831104","60397977600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","230","2512","25052","235224","2115432","18417024","156323088","1300080672","10632543840","85745329920","683292625344","5389506564480","42132573615744"],"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"],["2","16"]],"recurrence":{"denominator":{"coeffs":[["9","-60","16","128","32","-32"],["0","54","-126","-120","104","48"],["0","0","78","-44","-116","0"],["0","0","0","36","8","-16"],["0","0","0","0","4","0"]]},"terms":[{"coeff":{"coeffs":[["18","24","-88","-128","64","128","0"],["0","108","276","-80","-576","-352","0"],["0","0","156","440","120","-368","-192"],["0","0","0","72","192","64","-64"],["0","0","0","0","8","16","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["18","24","-88","-128","64","128","0"],["0","108","192","-352","-752","-288","0"],["0","0","156","368","-72","-432","-128"],["0","0","0","72","176","32","-64"],["0","0","0","0","8","16","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182125,"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-7dc57f28108bac44"]},{"id":182126,"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-7dc57f28108bac44"]},{"id":182127,"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-7dc57f28108bac44"]}],"region_exact":{},"incoming_edges":[{"id":"e-7dc57f28108bac44","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-2ea32e7d80305e8d","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":"28d6c7165a3dcaaef46d5fd72650f87bbd873874bc85362e4dcf892c719ba077","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-2156f740e0c97180","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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-c9f7d6b7b4e99504","kind":"fingerprint_related","node_a_id":"n-2ea32e7d80305e8d","node_b_id":"n-e16a5c5bc932c32c","fingerprint":"f6-5b508ded3910d61e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5b508ded3910d61e"},"other_id":"n-e16a5c5bc932c32c","other":{"id":"n-e16a5c5bc932c32c","name":"theta_plus_one_squared of basis_map_A013609 of Riordan array (1, 2+2x).","first_index":0,"fingerprint":"f6-5b508ded3910d61e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","16"],["6","80","144"]],"card_rows":[["1"],["2","16"],["6","80","144"],["16","320","1152","1024"]],"row_sums_preview":["1","18","230","2512","25052","235224","2115432","18417024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f5a001953ee28e8f","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-1a65135df7496975","e-28b2c657da30d993"],"text":"A099040 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A099040 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","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:    2,     16;\n   3:    6,     80,     144;\n   4:   16,    320,    1152,     1024;\n   5:   44,   1152,    6192,    11264,      6400;\n   6:  120,   3904,   27936,    76800,     89600,     36864;\n   7:  328,  12736,  113760,   418816,    742400,    626688,    200704;\n   8:  896,  40448,  433152,  1998848,   4761600,   6119424,   4014080,   1048576;\n   9: 2448, 125952, 1572480,  8724480,  26118400,  45195264,  45158400,  24117248,   5308416;\n  10: 6688, 386304, 5508864, 35708928, 128780800, 280498176, 377724928, 307232768, 138018816, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (9+(54*n-60)*x+(78*n^2-126*n+16)*x^2+(36*n^3-44*n^2-120*n+128)*x^3+(4*n^4+8*n^3-116*n^2+104*n+32)*x^4-(16*n^3-48*n+32)*x^5)*P(n,x) = (18+(108*n+24)*x+(156*n^2+276*n-88)*x^2+(72*n^3+440*n^2-80*n-128)*x^3+(8*n^4+192*n^3+120*n^2-576*n+64)*x^4+(16*n^4+64*n^3-368*n^2-352*n+128)*x^5-(64*n^3+192*n^2)*x^6)*P(n-2,x) + (18+(108*n+24)*x+(156*n^2+192*n-88)*x^2+(72*n^3+368*n^2-352*n-128)*x^3+(8*n^4+176*n^3-72*n^2-752*n+64)*x^4+(16*n^4+32*n^3-432*n^2-288*n+128)*x^5-(64*n^3+128*n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 16*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[((18 + (108*n + 24)*x + (156*n^2 + 276*n - 88)*x^2 + (72*n^3 + 440*n^2 - 80*n - 128)*x^3 + (8*n^4 + 192*n^3 + 120*n^2 - 576*n + 64)*x^4 + (16*n^4 + 64*n^3 - 368*n^2 - 352*n + 128)*x^5 - (64*n^3 + 192*n^2)*x^6)*P[n - 2, x] + (18 + (108*n + 24)*x + (156*n^2 + 192*n - 88)*x^2 + (72*n^3 + 368*n^2 - 352*n - 128)*x^3 + (8*n^4 + 176*n^3 - 72*n^2 - 752*n + 64)*x^4 + (16*n^4 + 32*n^3 - 432*n^2 - 288*n + 128)*x^5 - (64*n^3 + 128*n^2)*x^6)*P[n - 1, x])/(9 + (54*n - 60)*x + (78*n^2 - 126*n + 16)*x^2 + (36*n^3 - 44*n^2 - 120*n + 128)*x^3 + (4*n^4 + 8*n^3 - 116*n^2 + 104*n + 32)*x^4 - (16*n^3 - 48*n + 32)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 16*x; for(n = 3, N, my(i = n); v[i] = ((18 + (108*n + 24)*x + (156*n^2 + 276*n - 88)*x^2 + (72*n^3 + 440*n^2 - 80*n - 128)*x^3 + (8*n^4 + 192*n^3 + 120*n^2 - 576*n + 64)*x^4 + (16*n^4 + 64*n^3 - 368*n^2 - 352*n + 128)*x^5 - (64*n^3 + 192*n^2)*x^6)*v[i - 2] + (18 + (108*n + 24)*x + (156*n^2 + 192*n - 88)*x^2 + (72*n^3 + 368*n^2 - 352*n - 128)*x^3 + (8*n^4 + 176*n^3 - 72*n^2 - 752*n + 64)*x^4 + (16*n^4 + 32*n^3 - 432*n^2 - 288*n + 128)*x^5 - (64*n^3 + 128*n^2)*x^6)*v[i - 1])/(9 + (54*n - 60)*x + (78*n^2 - 126*n + 16)*x^2 + (36*n^3 - 44*n^2 - 120*n + 128)*x^3 + (4*n^4 + 8*n^3 - 116*n^2 + 104*n + 32)*x^4 - (16*n^3 - 48*n + 32)*x^5)); 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-c72653d9ab0ee8be","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-13a04d0165daacbb","e-7dc57f28108bac44"],"text":"A063967 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A063967 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}
