{"id":"n-ce43f3f34cdc34f8","revision":1,"first_index":1,"fingerprint":"f6-4ef688dcc1f7919d","name":"obreschkoff_combination of Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.","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"],["0","2","3"],["0","2","13","15"],["0","2","35","120","105"],["0","2","81","595","1365","945"],["0","2","175","2408","10710","18270","10395"],["0","2","365","8743","66430","207900","280665","135135"],["0","2","747","29760","359975","1827210","4369365","4864860","2027025"],["0","2","1513","97251","1790195","13758745","51606555","99324225","93918825","34459425"],["0","2","3047","309408","8402108","93553460","514383870","1516875360","2435132700","1998646650","654729075"],["0","2","6117","966823","37865828","592311720","4571036470","19315145850","46690223580","64163449350","46485764325","13749310575"],["0","2","12259","2984072","165730565","3560496940","37414086710","217017801000","739087278930","1508932274850","1810325892375","1173274502400","316234143225"],["0","2","24545","9132019","709935681","20588953385","288062584810","2225888094430","10235161386450","29056876348500","51237569708325","54497684015775","31939648465725","7905853580625"],["0","2","49119","27780600","2992734706","115588829538","2116516006605","21309158430560","128425816138620","485990697342150","1179237120086025","1827411702316200","1744558356791250","932890722513750","213458046676875"],["0","2","98269","84160455","12464109946","634230951068","14993333293575","193356192494465","1494181251422860","7318169626177410","23428537793736075","49532983652777625","68394066382892250","59199031611720000","29101447030280625","6190283353629375"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["2","5","30","262","2988","41960","699240","13478944","294956736","7222035680","195606080640","5806015203328","187408052230272","6535516745696000","244867697007744000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1","1"],["0","2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-2"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162790,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-0730b0b2fd7e59bf"]}],"region_exact":{},"incoming_edges":[{"id":"e-0730b0b2fd7e59bf","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-ce43f3f34cdc34f8","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"d7ebffe45bf2038eee57525dd6764876e705047f77a1506154f9e4a71aa03fea","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-eda3c556a6b4857f","role":"sequence","name":"Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-eda3c556a6b4857f","role":"sequence","name":"Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-a2167e29e1bc1f47","kind":"fingerprint_related","node_a_id":"n-ce43f3f34cdc34f8","node_b_id":"n-dffea903f52b4d20","fingerprint":"f6-4ef688dcc1f7919d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4ef688dcc1f7919d"},"other_id":"n-dffea903f52b4d20","other":{"id":"n-dffea903f52b4d20","name":"obreschkoff_combination of Triangle of Ward numbers T(n,k) read by rows.","first_index":1,"fingerprint":"f6-4ef688dcc1f7919d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2","3"],["2","13","15"],["2","35","120","105"]],"card_rows":[["2","3"],["2","13","15"],["2","35","120","105"],["2","81","595","1365","945"]],"row_sums_preview":["5","30","262","2988","41960","699240","13478944","294956736"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3fea9ad1525b48eb","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-72ea2d9fc8c7fc63"],"text":"A134991 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A134991 \u2192 P\u2099 + P\u2099\u208a\u2081","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, 1;\n   2: 0, 2,    3;\n   3: 0, 2,   13,     15;\n   4: 0, 2,   35,    120,     105;\n   5: 0, 2,   81,    595,    1365,      945;\n   6: 0, 2,  175,   2408,   10710,    18270,     10395;\n   7: 0, 2,  365,   8743,   66430,   207900,    280665,     135135;\n   8: 0, 2,  747,  29760,  359975,  1827210,   4369365,    4864860,    2027025;\n   9: 0, 2, 1513,  97251, 1790195, 13758745,  51606555,   99324225,   93918825,   34459425;\n  10: 0, 2, 3047, 309408, 8402108, 93553460, 514383870, 1516875360, 2435132700, 1998646650, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-2)*x*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) - (1-n*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + x, P(2,x) = 2*x + 3*x^2.","mathematica":"P[1, x_] := 1 + x;\nP[2, x_] := 2*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[(n - 2)*x*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x] - (1 - n*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + x; v[2] = 2*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (n - 2)*x*v[i - 2] + (x + x^2)*deriv(v[i - 2], x) - (1 - n*x)*v[i - 1] + (x + x^2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-eda3c556a6b4857f","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0730b0b2fd7e59bf"],"text":"A269939 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A269939 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
