{"id":"n-75ad3144101df727","revision":0,"first_index":0,"fingerprint":"f6-c7f8d68687ddb27c","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","4"],["2","12","12"],["2","20","48","32"],["4","48","156","192","80"],["4","64","300","608","560","192"],["8","144","792","2016","2640","1728","448"],["8","176","1224","4128","7680","8064","4480","1024"],["16","384","2976","11520","25680","34560","27776","12288","2304"],["16","448","4128","19456","54480","96192","108416","75776","29952","5120"],["32","960","9600","49920","157600","323136","441280","399360","230400","76800","11264"],["32","1088","12480","75520","282400","701376","1195264","1408000","1128960","588800","180224","24576"],["64","2304","28224","184320","753600","2080512","4027072","5548032","5425920","3686400","1655808","442368","53248"],["64","2560","35136","259584","1214400","3889152","8881600","14752768","17908992","15744000","9765888","4055040","1011712","114688"],["128","5376","77952","612864","3077760","10692864","26837888","49806336","69011712","71285760","54168576","29417472","10809344","2408448","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","102","480","1728","7776","26784","117504","393984","1700352","5598720","23887872","77635584","328458240"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["8","26","20","0","0"],["0","12","29","4","-16"],["0","0","9","24","16"]]},"terms":[{"coeff":{"coeffs":[["32","172","304","148","-88","-64"],["-24","-48","128","402","296","32"],["0","-54","-195","-182","48","96"],["0","0","-27","-108","-144","-64"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","52","116","80","-56","-96","-32"],["0","24","146","326","316","112","0"],["0","0","18","102","212","192","64"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174917,"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-81028b4ee75cb61c"]},{"id":174918,"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-81028b4ee75cb61c"]},{"id":174919,"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-81028b4ee75cb61c"]}],"region_exact":{},"incoming_edges":[{"id":"e-81028b4ee75cb61c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-75ad3144101df727","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":"d7792d87651fa0866b24e19e75aecae2c1707c44d783d759e517a451a50a0d31","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-e42a65125928b4c5","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.","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":[{"id":"l-e74990eb81a6c2b5","kind":"fingerprint_related","node_a_id":"n-ad6cd90b76531a4c","node_b_id":"n-75ad3144101df727","fingerprint":"f6-c7f8d68687ddb27c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c7f8d68687ddb27c"},"other_id":"n-ad6cd90b76531a4c","other":{"id":"n-ad6cd90b76531a4c","name":"theta_plus_one of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","first_index":0,"fingerprint":"f6-c7f8d68687ddb27c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1","6","6"],["1","10","24","16"],["1","12","39","48","20"]],"card_rows":[["1","6","6"],["1","10","24","16"],["1","12","39","48","20"],["1","16","75","152","140","48"]],"row_sums_preview":["13","51","120","432","972","3348","7344","24624"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5a70a6e97131d437","steps":["\u03b8+1"],"edges":["e-cb26430ba269e72d"],"text":"A064861 \u2192 \u03b8+1"},"headline":"A064861 \u2192 \u03b8+1","chain":null}}],"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:  1;\n  1:  1,   4;\n  2:  2,  12,   12;\n  3:  2,  20,   48,    32;\n  4:  4,  48,  156,   192,    80;\n  5:  4,  64,  300,   608,   560,   192;\n  6:  8, 144,  792,  2016,  2640,  1728,    448;\n  7:  8, 176, 1224,  4128,  7680,  8064,   4480,  1024;\n  8: 16, 384, 2976, 11520, 25680, 34560,  27776, 12288,  2304;\n  9: 16, 448, 4128, 19456, 54480, 96192, 108416, 75776, 29952, 5120;\n","recurrence_sentence":"The row polynomials satisfy (8+(12*n+26)*x+(9*n^2+29*n+20)*x^2+(24*n^2+4*n)*x^3+(16*n^2-16*n)*x^4)*P(n,x) = -(24*n-32+(54*n^2+48*n-172)*x+(27*n^3+195*n^2-128*n-304)*x^2+(108*n^3+182*n^2-402*n-148)*x^3+(144*n^3-48*n^2-296*n+88)*x^4+(64*n^3-96*n^2-32*n+64)*x^5)*P(n-1,x) + (8+(24*n+52)*x+(18*n^2+146*n+116)*x^2+(102*n^2+326*n+80)*x^3+(212*n^2+316*n-56)*x^4+(192*n^2+112*n-96)*x^5+(64*n^2-32)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(24*n - 32 + (54*n^2 + 48*n - 172)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (108*n^3 + 182*n^2 - 402*n - 148)*x^3 + (144*n^3 - 48*n^2 - 296*n + 88)*x^4 + (64*n^3 - 96*n^2 - 32*n + 64)*x^5)*P[n - 1, x] + (8 + (24*n + 52)*x + (18*n^2 + 146*n + 116)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (212*n^2 + 316*n - 56)*x^4 + (192*n^2 + 112*n - 96)*x^5 + (64*n^2 - 32)*x^6)*D[P[n - 1, x], x])/(8 + (12*n + 26)*x + (9*n^2 + 29*n + 20)*x^2 + (24*n^2 + 4*n)*x^3 + (16*n^2 - 16*n)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(24*n - 32 + (54*n^2 + 48*n - 172)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (108*n^3 + 182*n^2 - 402*n - 148)*x^3 + (144*n^3 - 48*n^2 - 296*n + 88)*x^4 + (64*n^3 - 96*n^2 - 32*n + 64)*x^5)*v[i - 1] + (8 + (24*n + 52)*x + (18*n^2 + 146*n + 116)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (212*n^2 + 316*n - 56)*x^4 + (192*n^2 + 112*n - 96)*x^5 + (64*n^2 - 32)*x^6)*deriv(v[i - 1], x))/(8 + (12*n + 26)*x + (9*n^2 + 29*n + 20)*x^2 + (24*n^2 + 4*n)*x^3 + (16*n^2 - 16*n)*x^4)); 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-35bd700768cac55f","steps":["basis A013609","\u03b8+1"],"edges":["e-69752228c1d1e91e","e-81028b4ee75cb61c"],"text":"A103633 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A103633 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}