{"id":"n-947462a675f7df3c","revision":0,"first_index":1,"fingerprint":"f6-de694c98c2b9f251","name":"theta_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= 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"],["2","8"],["5","40","48"],["16","184","432","256"],["65","928","3120","3584","1280"],["326","5352","22560","37120","25600","6144"],["1957","35432","174432","362240","358400","165888","28672"],["13700","267000","1472688","3605504","4524800","3010560","1003520","131072"],["109601","2264576","13635168","37768192","56241920","47652864","22880256","5767168","589824"],["986410","21386568","138157632","421558784","715061760","722405376","443154432","161218560","31850496","2621440"],["9864101","222762344","1525336848","5034073600","9467296000","10949879808","8040001536","3747348480","1070530560","170393600","11534336"],["108505112","2538195256","18259039920","64328425216","131627302400","169617524736","143147913216","80110288896","29390929920","6776422400","888143872","50331648"],["1302061345","31414193440","235817985168","878276264448","1928656801280","2716390195200","2564031303680","1654603382784","730802552832","216845516800","41235251200","4529848320","218103808"],["16926797486","419756855848","3270828473952","12781622520576","29813048911360","45251605082112","46855150870528","33924409851904","17324259803136","6202531512320","1520756064256","242749538304","22682796032","939524096"],["236975164805","6023382702888","48515334437184","197751164898560","486031307856640","786563638468608","880669013159936","701551279210496","403023308193792","167215691202560","49566421352448","10218381508608","1389321256960","111803367424","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","93","888","8977","97102","1127021","14028844","186909569","2658401458","40249021213","646843022592","11004123460305","197647268601910","3738871049312749"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","8"],["5","40","48"],["16","184","432","256"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-12"],["7"],["-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["9","-24"],["-9","8"],["2","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["4","8","-16"],["0","-8","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","8"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184822,"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-d39cec92ef158e3f"]},{"id":184823,"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-d39cec92ef158e3f"]},{"id":184824,"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-d39cec92ef158e3f"]},{"id":184825,"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-d39cec92ef158e3f"]}],"region_exact":{},"incoming_edges":[{"id":"e-d39cec92ef158e3f","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-947462a675f7df3c","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":"f7b6301feafb047f6d3c5578e556fd3ac6c63b21432f161519deb6546c03c32a","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-735a959cc5ec274b","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= 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:      2,        8;\n   3:      5,       40,        48;\n   4:     16,      184,       432,       256;\n   5:     65,      928,      3120,      3584,      1280;\n   6:    326,     5352,     22560,     37120,     25600,      6144;\n   7:   1957,    35432,    174432,    362240,    358400,    165888,     28672;\n   8:  13700,   267000,   1472688,   3605504,   4524800,   3010560,   1003520,    131072;\n   9: 109601,  2264576,  13635168,  37768192,  56241920,  47652864,  22880256,   5767168,   589824;\n  10: 986410, 21386568, 138157632, 421558784, 715061760, 722405376, 443154432, 161218560, 31850496, 2621440;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(8*n-24)*x)*P(n-3,x) - (n^2-4+(8*n-8)*x+16*x^2)*P(n-2,x) + (2*n-1+8*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 8*x, P(3,x) = 5 + 40*x + 48*x^2, P(4,x) = 16 + 184*x + 432*x^2 + 256*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 8*x;\nP[3, x_] := 5 + 40*x + 48*x^2;\nP[4, x_] := 16 + 184*x + 432*x^2 + 256*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*P[n - 3, x] - (n^2 - 4 + (8*n - 8)*x + 16*x^2)*P[n - 2, x] + (2*n - 1 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 8*x; v[3] = 5 + 40*x + 48*x^2; v[4] = 16 + 184*x + 432*x^2 + 256*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*v[i - 3] - (n^2 - 4 + (8*n - 8)*x + 16*x^2)*v[i - 2] + (2*n - 1 + 8*x)*v[i - 1]); 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-3272d5bc697b9cb4","steps":["basis A013611","\u03b8+1"],"edges":["e-91983714a64825b5","e-d39cec92ef158e3f"],"text":"A094816 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A094816 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}