{"id":"n-05ff6d2347b5ee68","revision":0,"first_index":0,"fingerprint":"f6-73b18cc810b5ad72","name":"basis_map_A008277 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","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"],["10","2"],["100","44","4"],["1000","728","144","8"],["10000","10736","3472","416","16"],["100000","148832","70080","13600","1120","32"],["1000000","1985984","1278784","357760","47360","2880","64"],["10000000","25831808","21874944","8281728","1568000","152320","7168","128"],["100000000","329981696","357912832","176257536","44787456","6182400","462336","17408","256"],["1000000000","4159780352","5670743040","3535946240","1158689280","213222912","22536192","1342464","41472","512"],["10000000000","51917364224","87709963264","67916625920","27928299520","6581836800","922242048","77291520","3763200","97280","1024"],["100000000000","643008370688","1331774214144","1262085941248","638542643200","187493335040","33452998656","3699480576","252426240","10250240","225280","2048"],["1000000000000","7916100448256","19930855739392","22856923488256","14017939460096","5026951987200","1110952640512","155693531136","13962043392","791859200","27258880","516096","4096"],["10000000000000","96993205379072","294864181248000","405572487290880","298036757258240","128574918664192","34494862065664","5958550028288","674400190464","50096144384","2401484800","71032832","1171456","8192"],["100000000000000","1183918464548864","4322084948230144","7078888159150080","6175806605230080","3167571887800320","1016036802772992","211994924810240","29451505008640","2751492423680","172236832768","7076020224","181895168","2637824","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","148","1880","24640","333664","4672832","67716096","1015601920","15762302464","253057484800","4200319887360","72030198976512","1275221931966464","23288684287377408"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["10","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197491,"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-db9683720c75da67"]},{"id":197492,"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-db9683720c75da67"]},{"id":197493,"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-db9683720c75da67"]}],"region_exact":{},"incoming_edges":[{"id":"e-db9683720c75da67","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-05ff6d2347b5ee68","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"a92070a49620e7207583443a9cd5eb5f2f6e9574430ebb0876cdf0190176fb9a","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-a4775661709ca859","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","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-013620ba69b0a542","kind":"fingerprint_related","node_a_id":"n-05ff6d2347b5ee68","node_b_id":"n-b1ad4f9d2fc1e443","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-b1ad4f9d2fc1e443","other":{"id":"n-b1ad4f9d2fc1e443","name":"basis_map_A008277 of basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["20","4"],["400","176","16"]],"card_rows":[["1"],["20","4"],["400","176","16"],["8000","5824","1152","64"]],"row_sums_preview":["1","24","592","15040","394240","10677248","299061248","8667660288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611","basis A008277"],"edges":["e-9c08db68889738c4","e-f32fbf019a97b46e"],"text":"A027467 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A027467 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-1d859abb6efa4208","kind":"fingerprint_related","node_a_id":"n-bac12d13055658fe","node_b_id":"n-05ff6d2347b5ee68","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-bac12d13055658fe","other":{"id":"n-bac12d13055658fe","name":"5-Stirling numbers of the second kind.","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","1"],["25","11","1"]],"card_rows":[["1"],["5","1"],["25","11","1"],["125","91","18","1"]],"row_sums_preview":["1","6","37","235","1540","10427","73013","529032"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A193685"],"derivation":{"root":"n-bac12d13055658fe","steps":[],"edges":[],"text":"A193685"},"headline":"A193685: 5-Stirling numbers of the second kind.","chain":null}},{"id":"l-5854fa608b9bb808","kind":"fingerprint_related","node_a_id":"n-802eabe88630cab9","node_b_id":"n-05ff6d2347b5ee68","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-802eabe88630cab9","other":{"id":"n-802eabe88630cab9","name":"reversal of 5-Stirling numbers of the second kind.","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","5"],["1","11","25"]],"card_rows":[["1"],["1","5"],["1","11","25"],["1","18","91","125"]],"row_sums_preview":["1","6","37","235","1540","10427","73013","529032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bac12d13055658fe","steps":["reverse"],"edges":["e-1fda128e00cfeef7"],"text":"A193685 \u2192 reverse"},"headline":"A193685 \u2192 reverse","chain":null}},{"id":"l-8e3e529772bde093","kind":"fingerprint_related","node_a_id":"n-67748ed3490e4da5","node_b_id":"n-05ff6d2347b5ee68","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-67748ed3490e4da5","other":{"id":"n-67748ed3490e4da5","name":"one_plus_derivative of one_plus_derivative of Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","1"],["25","11","1"]],"card_rows":[["1"],["5","1"],["25","11","1"],["125","91","18","1"]],"row_sums_preview":["1","6","37","235","1540","10427","73013","529032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["D+1","D+1"],"edges":["e-71c634e76c593d53","e-df31745c2eccb007"],"text":"A143495 \u2192 D+1 \u2192 D+1"},"headline":"A143495 \u2192 D+1 \u2192 D+1","chain":null}},{"id":"l-c41b9235e8f451b5","kind":"fingerprint_related","node_a_id":"n-05ff6d2347b5ee68","node_b_id":"n-19d88b15847dfa77","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-19d88b15847dfa77","other":{"id":"n-19d88b15847dfa77","name":"basis_map_A008277 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["400","176","16"],["8000","5824","1152","64"],["160000","171776","55552","6656","256"]],"card_rows":[["400","176","16"],["8000","5824","1152","64"],["160000","171776","55552","6656","256"],["3200000","4762624","2242560","435200","35840","1024"]],"row_sums_preview":["592","15040","394240","10677248","299061248","8667660288","259994091520","8070298861568"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318","basis A008277"],"edges":["e-7b68ca495e98f7ea","e-5a0ab736f64a7d9b"],"text":"A038330 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A038330 \u2192 basis A007318 \u2192 basis A008277","chain":null}},{"id":"l-cff201f312825e1c","kind":"fingerprint_related","node_a_id":"n-8fc0bee0ec2b787b","node_b_id":"n-05ff6d2347b5ee68","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-8fc0bee0ec2b787b","other":{"id":"n-8fc0bee0ec2b787b","name":"basis_map_A008277 of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["10","2"]],"card_rows":[["1"],["10","2"],["75","33","3"],["500","364","72","4"]],"row_sums_preview":["0","1","12","111","940","7700","62562","511091"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D","basis A008277"],"edges":["e-62aaaa59f91d42c1","e-c9d2a137ac16a4ae"],"text":"A038231 \u2192 D \u2192 basis A008277"},"headline":"A038231 \u2192 D \u2192 basis A008277","chain":null}},{"id":"l-db55d077ce2d6189","kind":"fingerprint_related","node_a_id":"n-f0b949b616a94e3a","node_b_id":"n-05ff6d2347b5ee68","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-f0b949b616a94e3a","other":{"id":"n-f0b949b616a94e3a","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["225","99","9"],["3375","2457","486","27"],["50625","54351","17577","2106","81"]],"card_rows":[["225","99","9"],["3375","2457","486","27"],["50625","54351","17577","2106","81"],["759375","1130193","532170","103275","8505","243"]],"row_sums_preview":["333","6345","124740","2533761","53226477","1156992984","26028766395","605955858201"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A008277"],"edges":["e-f14e261f691e9869"],"text":"A038329 \u2192 basis A008277"},"headline":"A038329 \u2192 basis A008277","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:         10,          2;\n  2:        100,         44,          4;\n  3:       1000,        728,        144,          8;\n  4:      10000,      10736,       3472,        416,         16;\n  5:     100000,     148832,      70080,      13600,       1120,        32;\n  6:    1000000,    1985984,    1278784,     357760,      47360,      2880,       64;\n  7:   10000000,   25831808,   21874944,    8281728,    1568000,    152320,     7168,     128;\n  8:  100000000,  329981696,  357912832,  176257536,   44787456,   6182400,   462336,   17408,   256;\n  9: 1000000000, 4159780352, 5670743040, 3535946240, 1158689280, 213222912, 22536192, 1342464, 41472, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (10+2*x)*P(n-1,x) + 2*x*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[(10 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\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] = (10 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 1], x)); 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-f211cb6153c37c6d","steps":["basis A013609","basis A008277"],"edges":["e-13f16c9d95bb60f6","e-db9683720c75da67"],"text":"A027466 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A027466 \u2192 basis A013609 \u2192 basis A008277","chain":null}
