{"id":"n-ca40f10a3c367093","revision":0,"first_index":1,"fingerprint":"f6-7c3b635378939853","name":"theta_plus_one of Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.","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":[["4","24","12"],["8","104","156","32"],["16","392","1248","784","80"],["32","1368","7932","10576","3420","192"],["64","4552","44040","105760","73400","13656","448"],["128","14680","223956","881120","1101400","447912","51380","1024"],["256","46344","1071984","6495856","13216800","9743784","2501296","185376","2304"],["512","144152","4913580","43913776","136139540","163367448","76849108","13102880","648684","5120"],["1024","443720","21816600","278531840","1255975000","2287144272","1758365000","557063680","65449800","2218600","11264"],["2048","1355736","94587780","1683547200","10669333200","28067369904","32745264888","17070933120","3787981200","315292600","7456548","24576"],["4096","4120456","402754368","9805023440","85060339200","311700387888","523924238208","415600517184","153108610560","24512558600","1476766016","24722736","53248"],["8192","12476056","1691366364","55469187088","645181107500","3202626785616","7464247978488","8530569118272","4803940178424","1290362215000","152540264492","6765465456","81094364","114688"],["16384","37673928","7027462632","306662952416","4703124451320","30934778118312","97077962993424","153550244128896","124814523848688","51557963530520","10346873792904","919988857248","30452338072","263717496","245760"],["32768","113546072","28957513908","1664494064544","33202019684680","284268438927192","1173620023132396","2491414631370624","2802841460291952","1676600033046280","521158804699852","79684847243232","5409605709768","135135064904","851595540","524288"],["65536","341752328","118555161360","8901620600880","228340764237600","2507830579571352","13368718330039856","37176811973142880","56056829205839040","46471014966428600","21007985947205488","5015661159142704","593685987017760","31155672103080","592775806800","2734018624","1114112"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["40","300","2520","23520","241920","2721600","33264000","439084800","6227020800","94443148800","1525620096000","26153487360000","474249904128000","9070029416448000","182467650613248000"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["4","24","12"],["8","104","156","32"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-6","-6","-6"],["-2","-8","-11"],["0","-2","-6"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-3","6"],["0","-1","-4","5"],["0","0","-1","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","8"],["3","8"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"],["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188124,"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-995d7d6de5709b6d"]},{"id":188125,"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-995d7d6de5709b6d"]},{"id":188126,"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-995d7d6de5709b6d"]},{"id":188127,"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-995d7d6de5709b6d"]}],"region_exact":{},"incoming_edges":[{"id":"e-995d7d6de5709b6d","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-ca40f10a3c367093","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":"37a3a3896390d9292d86c40274bd92d50ed6f551d5c14886cf8bd25c806f64eb","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-9c8a3637306e75dc","role":"sequence","name":"Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.","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-0c6072b1b75fabcf","kind":"fingerprint_related","node_a_id":"n-ca40f10a3c367093","node_b_id":"n-5ed42a17b514c663","fingerprint":"f6-7c3b635378939853","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7c3b635378939853"},"other_id":"n-5ed42a17b514c663","other":{"id":"n-5ed42a17b514c663","name":"theta_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","first_index":0,"fingerprint":"f6-7c3b635378939853","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","12"],["36","216","108"]],"card_rows":[["1"],["6","12"],["36","216","108"],["216","2808","4212","864"]],"row_sums_preview":["1","18","360","8100","204120","5715360","176359680","5952139200"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-61cf3a32da4a0e22","steps":["\u03b8+1"],"edges":["e-fc5203ebe83181fc"],"text":"A257626 \u2192 \u03b8+1"},"headline":"A257626 \u2192 \u03b8+1","chain":null}},{"id":"l-d5597631237d6f0d","kind":"fingerprint_related","node_a_id":"n-ca40f10a3c367093","node_b_id":"n-db27be2f5676b05b","fingerprint":"f6-7c3b635378939853","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7c3b635378939853"},"other_id":"n-db27be2f5676b05b","other":{"id":"n-db27be2f5676b05b","name":"theta_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.","first_index":0,"fingerprint":"f6-7c3b635378939853","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","8"],["16","96","48"]],"card_rows":[["1"],["4","8"],["16","96","48"],["64","832","1248","256"]],"row_sums_preview":["1","12","160","2400","40320","752640","15482880","348364800"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c35e92273dac3e06","steps":["\u03b8+1"],"edges":["e-081dfcccae930628"],"text":"A257613 \u2192 \u03b8+1"},"headline":"A257613 \u2192 \u03b8+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:    4,      24,       12;\n   2:    8,     104,      156,         32;\n   3:   16,     392,     1248,        784,          80;\n   4:   32,    1368,     7932,      10576,        3420,         192;\n   5:   64,    4552,    44040,     105760,       73400,       13656,         448;\n   6:  128,   14680,   223956,     881120,     1101400,      447912,       51380,        1024;\n   7:  256,   46344,  1071984,    6495856,    13216800,     9743784,     2501296,      185376,       2304;\n   8:  512,  144152,  4913580,   43913776,   136139540,   163367448,    76849108,    13102880,     648684,      5120;\n   9: 1024,  443720, 21816600,  278531840,  1255975000,  2287144272,  1758365000,   557063680,   65449800,   2218600,   11264;\n  10: 2048, 1355736, 94587780, 1683547200, 10669333200, 28067369904, 32745264888, 17070933120, 3787981200, 315292600, 7456548, 24576;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -(2*n+6+(2*n^2+8*n+6)*x+(n^3+6*n^2+11*n+6)*x^2)*P(n-2,x) - ((n+3)*x+(n^2+4*n+3)*x^2-(n^2+5*n+6)*x^3)*d/dx P(n-2,x) + (3*n+7+(2*n^2+8*n+8)*x)*P(n-1,x) + ((n+2)*x-(n+2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 4 + 24*x + 12*x^2, P(2,x) = 8 + 104*x + 156*x^2 + 32*x^3.","mathematica":"P[1, x_] := 4 + 24*x + 12*x^2;\nP[2, x_] := 8 + 104*x + 156*x^2 + 32*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 6 + (2*n^2 + 8*n + 6)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*P[n - 2, x] - ((n + 3)*x + (n^2 + 4*n + 3)*x^2 - (n^2 + 5*n + 6)*x^3)*D[P[n - 2, x], x] + (3*n + 7 + (2*n^2 + 8*n + 8)*x)*P[n - 1, x] + ((n + 2)*x - (n + 2)*x^2)*D[P[n - 1, x], x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 4 + 24*x + 12*x^2; v[2] = 8 + 104*x + 156*x^2 + 32*x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n + 6 + (2*n^2 + 8*n + 6)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*v[i - 2] - ((n + 3)*x + (n^2 + 4*n + 3)*x^2 - (n^2 + 5*n + 6)*x^3)*deriv(v[i - 2], x) + (3*n + 7 + (2*n^2 + 8*n + 8)*x)*v[i - 1] + ((n + 2)*x - (n + 2)*x^2)*deriv(v[i - 1], x))/(n + 2)); 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-9c8a3637306e75dc","steps":["\u03b8+1"],"edges":["e-995d7d6de5709b6d"],"text":"A256890 \u2192 \u03b8+1"},"headline":"A256890 \u2192 \u03b8+1","chain":null}